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1 Table of Contents Overview and Table of Contents 1-2 Number Sense and Operations 3-16 Data Analysis, Probability and Discrete Math 17-27 Patterns, Algebra and Functions 28-34 Geometry and Measurement 35-39 Mathematical Processes and Reasoning 40-47 Time Trials 48-57 Solutions 58-96 This study packet is a compilation of materials that have been used in a 5-session study group to help teachers prepare for the Middle School AEPA (37). There are 5 sections, one on each strand, with background information and sample problems. There is also a section with two time trials, mini tests to help you practice pacing for the test. Many teachers have indicated having trouble finishing the test in the 4 hours provided. For the time trials, you should be trying to complete the multiple choice section in approximately 20 minutes for 10 questions. The time for the full test is 4 hours for 100 multiple choice questions and 1 extended response question. The extended response will probably take around 30-40 minutes. Solutions to all of the sample problems as well as the time trials are provided in the last section of the handout. Additional short and long study groups will be offered throughout the year if you are interested in study with colleagues. The information will be sent to math representatives, instructional coaches, past study group participants, and posted in the communiqué. A study guide for the AEPA is available free of charge at the following link. http://www.aepa.nesinc.com/AZ_viewSG_opener.asp The questions from the study guide that are connected to each strand are indicated on the cover pages. Please don‟t hesitate to contact us if we can be of assistance to you as you study or if you have feedback about how the study materials could be improved. Christie McDougall & Christina Harmon TUSD Curriculum, Instruction and Technology Integration 2 Number Sense and Operations Questions 1-4 and related materials 3 Real Numbers Rational Numbers Real numbers that can be expressed as a ratio of two integers Ex. Integers Whole numbers and their opposites Ex. Whole numbers Zero and natural numbers Ex. Natural Numbers (Counting Numbers) Ex. 4 Properties of Radicals and Rationalizing a Denominator What It Means to Rationalize the Denominator In order that all of us doing math can compare answers, we agree upon a common conversation, or set of rules, concerning the form of the answers. For instance, we could easily agree that we would not leave an answer in the form of 3 + 4, but would write 7 instead. When the topic switches to that of radicals, those doing math have agreed that a RADICAL IN SIMPLE FORM will not (among other things) have a radical in the denominator of a fraction. We will all change the form so there is no radical in the denominator. Now a radical in the denominator will not be something as simple as . Instead, it will have a radicand that will not come out from under the radical sign like . Since is an irrational number, and we need to make it NOT irrational, the process of changing its form so it is no longer irrational is called RATIONALIZING THE DENOMINATOR. In an expression with radicals is in simplest form if the following are true: No perfect square factors other than 1 are in the radicand. No fractions are in the radicand. No radicals appear in the denominator of a fraction. You may need to apply the product and quotient properties to help you simplify radicals. Product Property The square root of a product equals the product of the square roots of the factors. ab = a b when a and b are positive numbers ex. 4 100 = 4 100 Quotient Property The square root of a quotient equals the quotient of the square roots of the numerator and denominator. a a 9 9 = when a and b are positive numbers ex. = b b 25 25 Examples of Rationalizing the Denominator: For the following problems the instruction is to rationalize the denominator, which means to write an equivalent expression for it that doesn't have any radicals in the denominator. Example 1: 5 Solution: Explanation: There is just one square root in the denominator, so this is case 1. So to get rid of it we just multiply numerator and denominator by it. The square root of 3 times the square root of 3 is 3 simply from the meaning of square root, you don't have to write it first as the square root of 9. Remember that the square root of 3 means the number you can square, that is multiply by itself, to get 3. So by multiplying the top and the bottom by the square root of 3 we get rid of the square root in the denominator. The numerator appears more complicated, and in fact the whole thing does in fact look more complicated after this 'simplification', but many times, for example, when you are adding and need to find common denominators, this sacrifice is worth it, because simplicity is more important in denominators than in numerators. Example 2: Solution: Explanation: Here we first need to split up the quotient into two separate radicals. Then just like the last one we multiply top and bottom by the square root that is downstairs (in the denominator), the square root of 6, to turn the square root of 6 into 6. The numerator again suffers from it. Here since there was a square root upstairs (in the numerator) it made it look nicer to pull the radicals together under one radical and multiply. Example 3: Solution: Explanation: This one is similar except that there is something multiplied by the radical in the denominator. Here we don't have to multiply by the whole denominator, just the radical part of it. The 5 just sits there and comes along for the ride. Another interesting thing that happened is that after we multiplied the square roots of two to get 2, we were able to simplify further, because that 2 canceled with the 2 that was in the numerator. 6 Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator. Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent. Step 2: Make sure all radicals are simplified. Step 3: Simplify the fraction if needed. Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical. Simplifying Radical Expressions – Practice 1. Simplify the expression 50 . 2. Simplify the expressions. 3 20 32 a. b. c. 4 4 50 3. Body Surface Area Physicians can approximate the Body Surface Area of an adult (in sq. meters) using an index called BSA where H is height in cm and W is weight in kg. HW Body Surface Area (BSA) = 3000 Find the BSA of a person who is 160 cm tall and weighs 50 kg. 7 Exploring Quadratic Equation The general form of a quadratic equation is where a is not equal to zero. Every quadratic equation has U-shaped graph called a parabola. The point in the middle where the two half of the parabola meet is called the vertex. Let‟s explore the affect of changing the a, b and c values on the shape and position of the parabola. We will be graphing each of the following sets of graphs on the same grid using different colors of colored pencils. In order to explore the changing of the coefficients (a and b) and the constant (c), we will change one of the parameters, graph, and observe. Exploration 1: Start by graphing the parent function, y = x2. (the value of a=1, b=0 and c=0) Now let‟s explore what happens when the a value (coefficient of the x2) changes. Plot the following equations on the same grid. y = 3x2 y = ½ x2 y = -3x2 y = -½ x2 Write a summary statement about the affect of changing the value of a on the shape and position of the graph. Exploration 2: Graph the parent function on another graph, y = x2. Now let‟s explore what happens when the c value (constant value) changes. Plot the following equations on the same grid. y = x2 + 1 y = x2 – 1 y = x2 + 3 y = x2 + -3 Write a summary statement about the affect of changing the value of the constant (c) on the shape and position of the graph. Exploration 3: Graph the parent function on another graph, y = x2. Now let‟s explore what happens when the b value (the coefficient of the x term) changes. Plot the following equations on the same grid. y = x2 + x y = x2 + 3x y = x2 - x y = x2 -3x Write a summary statement about the affect of changing the value of b on the shape and position of the graph. Using the information: Make some predictions about the shape and position of the following graph. At how many places do you think the graph will cross the x-axis? Graph the equation using substitution to check your predictions. y = -x2 + 2x + 5 8 Solving Quadratic Equations by the Quadratic Formula Quadratic Functions A quadratic function is a function that can be written in the standard form y = ax2 + bx + c, where a 0 Every quadratic function has a U-shaped graph called a parabola. If the leading coefficient a is positive the parabola opens up. If the leading coefficient is negative, the parabola opens down. The vertex is the lowest point of a parabola that opens up and the highest point of a parabola that opens down. b The vertex has an x-coordinate of . a The line passing through the vertex that divides the parabola into two symmetric parts is called the axis of symmetry. The two symmetric parts are mirror images of each other. To graph a quadratic function: Step 1: Find the x-coordinate of the vertex. Step 2: Make a table of values, using x-values to the right and left of the vertex. Step 3: Plot the points and connect them with a smooth curve to form a parabola. Quadratic functions are very useful because they frequently model real-life situations, height and distance of a shot put throw or the arc of water from a fountain. Solving Quadratic Functions The roots or solutions of quadratic equations can be found in several ways. To solve a quadratic function, rewrite it in the form of ax2 + bx + c = 0 (make y equal to zero or solve so that all of the terms are on one side of the equation). Algebraically Some simple quadratic equations can be solved algebraically. 1 2 4. x 8 2 9 Using a Graph The solutions, or roots, of ax2 + bx + c = 0 are the x-intercepts of the graph of the parabola. 5. x 2 x 2 x y Factoring To factor a quadratic expression means to write it as the product of two linear expressions. You know from the FOIL method that (x+p)(x+q) = x2 + (p+q)x + pq. So to factor x2 + bx + c, you need to find numbers p and q such that p + q = b AND pq = c. 6. Solve x 2 3x 4 0 . 10 Using Quadratic Equation The quadratic formula is another way to solve a quadratic equation. The quadratic formula can be used in all cases. The solutions of the quadratic equation ax2 + bx + c = 0 are when a 0 and b2 – 4ac 0 (the discriminant). The formula can be read as “x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a.” If the discriminant (b2 – 4ac) is positive, then the equation has two solutions. If the discriminant (b2 – 4ac) is zero, then the equation has one solution. If the discriminant (b2 – 4ac) is negative, then the equation has no real solutions. 7. Solve 2x2 - 4x – 3 = 0, and round your answer to two decimal places, if necessary. 8. x(x-2) = 4 Hint: Make sure to write the equation in the form of quadratic = zero. 11 Applications of Quadratic Equations Vertical Motion Models Object is dropped h = -16t2 + s Object is thrown h = -16t2 + vt + s h = height (feet) t = time in motion (seconds) s = initial height (feet) v = initial velocity (feet per second) In these models, the coefficient of t2 is one half the acceleration due to gravity. On the surface of the Earth, this acceleration is approximately 32 feet per second per second. Use the vertical motion model to find how long it will take for the object to reach the ground. 9. You drop keys from a window 30 feet above ground to your friend below. Your friend does not catch them. 10. A lacrosse player throws a ball upward from her playing stick with an initial height of 7 feet, at an initial speed of 90 feet per second. 12 Factors, GCF, and LCM Prime Factorization Breaking up a composite number into its prime factors can help you understand the number and compute with it. A composite number written as the product of prime numbers is called the prime factorization of the number. A couple of ways you might find the prime factorization are by using a factor tree or division. 11. Find the prime factorization of 132. 12. Find the prime factorization of 72. 13. Find the prime factorization of 105. Finding GCFs and LCMs by Prime Factoring Now we are ready to use prime factorizations to find GCFs and LCMs. Using the condition that we just obtained, what are trying to do for the GCF is put together the largest collection of factors in a prime factorization so that it is contained in both of the prime factorizations, and for the LCM we want the smallest prime factorization that both of them are contained in. If you have heard of intersections and unions, it is sort of like the intersection for the GCF and the union for the LCM. If you haven't heard of such things, it looks sort of like this: GCF The intersection of A and B. A B LCM The union of A and B. A B That is we want to put together everything they have in common for the GCF and throw everything together, but without duplication for the LCM. You might find that it helps you remember the methods for these simply to have these pictures in mind. I tend to have pictures sort of like this in my mind when I am thinking about them. 13 Greatest Common Factor A couple of ways to find the GCF include: 1. Listing the factors of the numbers and looking for the greatest factor that is common to both lists. 2. Finding a prime factorization of each number and then finding the prime factors that are in common and multiplying them. 14. Find the GCF of 18 and 30. 15. Find the GCF of 96 and 144. Least Common Multiple A couple of ways to find the LCM include: 1. List the non-zero multiples until you find a match. 2. Finding a prime factorization of each number, find the common factors, and multiply the common factors and the extra factors. 16. Find the LCM of 8, 9, and 12. 17. Find the LCM of 18 and 30. 14 Applications of LCM and GCF Solve each problem and determine whether each is an application of least common multiple or greatest common factor. What do you notice about the problem types? 18. The front wheel of a tricycle has a circumference of 54 inches and the back wheels have a circumference of 36 inches. If points P and Q are both touching the sidewalk when Jose starts to ride, when will P and Q first touch the sidewalk at the same time again? Which tire would wear out first? Explain. 19. Two of your children ride 2 wheel bikes to school. Karen has a bike with wheel size 18 inches. Maria has a bike with wheel size of 26 inches. Karen was very tired and could not keep up with Maria on their ride to school. Explain to Karen why she had a different experience than Maria. 20. There are three 4th Grades, A, B, and C, at the Local Elementary School. Room 4-A contains 12 children, Room 4-B contains 18, and Room 4-C contains 24 children. The 4th graders will get together to play a game in the middle of the day. The teachers want the children to be on teams with their own classmates but all teams need to be of equal size. Can you find the size of the teams so that everyone plays and each team contains children from the same room? 21. Ms. Nguyen‟s class purchased 45 pencils and 30 erasers to put in care packages. They want each package to contain the same number of pencils and the same number of erasers. What is the greatest number of packages they can make? 22. The class is having a party. The students have voted to have hot dogs that come in packages of 10, with buns that come in packages of 8, and ice cream cups that come in packs of 24. What is the least number of packages of hot dogs, buns, and ice cream that you would need to buy to have the same number of each? 15 23. Dr. Pascal studies the effects of light sources on houseplants. The number of plants in each class are 24, 30, 36 and 42. He wants to subdivide the classes into groups of the same size for the research project. What is the largest group size that will work in all four classes? 24. A city recycling crew was told that, beginning on April 1, it would pick up different types of plastics according to the schedule shown below. How often will all three types be picked up on the same day? Recycling Pick Up Every 5 days Every 9 days Every 6 days 25. Suppose Earth and Mars were aligned with the sun. Earth completes its orbit in 365 days and Mars completes its orbit in 687 days. When do both planets return to these same positions in their orbits? 16 Data Analysis, Probability and Discrete Math Questions 5-8 and related materials 17 Box and Whisker Plots Box-and-whisker plots are helpful in interpreting the distribution of data. We know that the median of a set of data separates the data into two equal parts. Data can be further separated into quartiles. The first quartile is the median of the lower part of the data. The second quartile is another name for the median of the entire set of data. The third quartile is the median of the upper part of the data. Quartiles separate the original set of data into four equal parts. Each of these parts contains one-fourth of the data. Constructing a box-and-whisker plot: The data: Math test scores 80, 75, 90, 95, 65, 65, 80, 85, 70, 100 Write the data in numerical order and find the first quartile, the median, the third quartile, the smallest value and the largest value. median = 80 first quartile = 70 third quartile = 90 smallest value = 65 largest value = 100 Place a circle beneath each of these values on a number line. Draw a box with ends through the points for the first and third quartiles. Then draw a vertical line through the box at the median point. Now, draw the whiskers (or lines) from each end of the box to the smallest and largest values. 18 Special Case: You may see a box-and-whisker plot, like the one below, which contains an asterisk. Sometimes there is ONE piece of data that falls well outside the range of the other values. This single piece of data is called an outlier. If the outlier is included in the whisker, readers may think that there are grades dispersed throughout the whole range from the first quartile to the outlier. Finding the median for box and whisker data sets – even number of data points and odd number of data points Odd number of data points Ex. 65, 66, 67, 68, 69, 70, 71 Since there are an odd number of data points, the median is the middle number or 68. 65, 66, 67, 68, 69, 70, 71 To figure out the values for Q1 and Q3, use the numbers to the left and right of the median but not the median. The value for Q1 would be 66 and the value for Q3 would be 70. Odd number of data points Ex. 65, 66, 67, 68, 69, 70, 71, 82 Since there are an even number of data points, the median is the mean of the middle two numbers or 68.5. 65, 66, 67, 68, 69, 70, 71, 72 To figure out the values for Q1 and Q3, use the numbers to the left and right of the median. Since the median is an average this time include the 68 and 69 in their respective sets. The value for Q1 would be 66.5 and the value for Q3 would be 70.5. 19 Five Number Summary Maximum value (upper extreme) Quartile 3 (upper quartile) Median (Quartile 2) Quartile 1 (lower quartile) Minimum value (lower extreme) Quartiles In descriptive statistics, a quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents 1/4th of the sample or population. Thus: first quartile (designated Q1) = lower quartile = cuts off lowest 25% of data = 25th percentile second quartile (designated Q2) = median = cuts data set in half = 50th percentile third quartile (designated Q3) = upper quartile = cuts off highest 25% of data, or lowest 75% = 75th percentile The difference between the upper and lower quartiles is called the interquartile range. Percentiles One of a set of points on a scale arrived at by dividing a group into parts in order of magnitude. For example, a score equal to or greater than 97 percent of those attained on an examination is said to be in the 97th percentile. The percentile of an item can be calculated by the formula B Pr x100 N where Pr is the percentile, B is the number of items below the item, and N is the number of items in the data set. As related to a normal distribution: a percentile greater than 75 is considered above normal a percentile between 25 and 75 is considered normal a percentile less than 25 is considered below normal 20 Normal Distribution – mean and standard deviation A probability distribution shaped like a bell, often found in statistical samples. The distribution of the curve implies that for a large population of independent random numbers, the majority of the population often cluster near a central value, and the frequency of higher and lower values taper off smoothly. (ex. a person‟s height, intelligence as measured by an IQ test, scale used in norm reference tests) The Standard Normal curve, shown here, has mean 0 and standard deviation 1. If a dataset follows a normal distribution, then about 68% of the observations will fall within of the mean , which in this case is with the interval (-1,1). About 95% of the observations will fall within 2 standard deviations of the mean, which is the interval (-2,2) for the standard normal, and about 99.7% of the observations will fall within 3 standard deviations of the mean, which corresponds to the interval (-3,3) in this case. Although it may appear as if a normal distribution does not include any values beyond a certain interval, the density is actually positive for all values, . Data from any normal distribution may be transformed into data following the standard normal distribution by subtracting the mean and dividing by the standard deviation . Example Question from the GRE 1. Which is greater? (A) The standard deviation of the data 5, 5, 8, 14 and 18. (B) The standard deviation of the data 6, 6, 8, 14 and 16. The definition of Standard deviation is "the average distance from the arithmetic mean for the N measurements". So my study guide says. So if I apply this concept to the question above, first I must find the arithmetic mean of both data which is 10 for both (the sum of 5,5,8,14,18 divided by 6 is 10 for A; the same for B..). After this, what do I need to find out the standard deviation of (A) and (B)? Solution: At times, you need not do huge calculations to arrive at an answer. Standard deviation is a measure of spread (i.e. how the disperse the data is). Looking at A and B, we can see that B is more tightly packed than A, so the standard deviation for A necessarily be GREATER. We need not calculate the value of the Standard Deviation, we were asked only to rank them, so time calculating it would serve only to verify what we can do by inspection. Remember, in GRE like SAT, There are many ways to arrive at a solution, the key to testing well, is to step back, analyze the different options available to solve the problem, and choose the one that is most suited. Your testing time will decrease considerably and your confidence increased exponentially. 21 Additional Standard Deviation Questions 26. A normally distributed set of data has a mean of 20 and a standard deviation of 3. Determine the probability that a randomly selected x-value is between 14 and 26. 27. A normally distributed set of data has a mean of 60 and a standard deviation of 10. What percent of data fall between 50 and 70? 28. In a research project, it was found that the heights of coconut trees are normally distributed with a mean of 60 feet and a standard deviation of 5 feet. Determine the probability that five randomly selected coconut trees are all between the heights of 50 and 70 feet. 22 Scatterplots Scatter plots are similar to line graphs in that they use horizontal and vertical axes to plot data points. However, they have a very specific purpose. Scatter plots show how much one variable is affected by another. The relationship between two variables is called their correlation. Scatter plots usually consist of a large body of data. The closer the data points come when plotted to making a straight line, the higher the correlation between the two variables, or the stronger the relationship. Positive Correlation Examples of Positive Correlation: As summer heat increases, the electric bill increases As height increases, shoe size increases As there are more cars on the road, pollution increases As more years pass, the population increases As the neighborhood expands, there are more students in a school As there are more people in a room, the temperature in the room increases As there are less resources, there are less people living in an area (mining communities) As attendance rate is higher, student grades and AIMS achievement is higher As more time passes, the plant grows taller As your height increases, your arm span increases No Correlation Examples of No Correlation: Adult age vs. adult height Teacher‟s age and number of student in their class Length of nose vs. number of lies you tell Hair length vs. height Height vs. how you did on your last test Birthdate vs. birth weight The number of letters in your first name vs. the number of letters in your last name The day vs. month of your birthday 23 Negative Correlation Examples of Negative Correlation: The higher the population, the less resources there are The lower the interest rate, the more houses that are sold The longer one lives the lower the value of the dollar The higher the altitude, the lower the temperature The further you travel, the less gas will be in the tank The more money spent, the less money saved The faster the speed, the quicker the time The higher the level of education, the fewer children in families The taller the building, the deeper the foundation The more HW missed, the lower the exam score The higher fuel prices are, the fewer airline flights As the performance of stocks increases, the performance of bonds decreases As you grow older, the rate of normal vision declines Tax rate vs. Unemployment rate SAT scores increase as % of seniors taking the test decrease As the minutes of exercise per week increases, resting heart rate decreases As the number of drinks ingested increase, the manual dexterity score decreases As more cigarettes are smoked, the birth weight of a child decreases Infant mortality tend to be higher in countries with a lower GNP per capital As your age now increases, the number of years until you are 60 decreases As automobile weight increases, fuel efficiency decreases As age at first word increases, the child‟s Gessel score decreases 24 Probability Summary The probability of an event is the number of ways it can occur, divided by the total number of possible outcomes. Probability is always a number between 0 (no outcomes lead to the event) and 1(all outcomes lead to the event.) Probability can be expressed as a percent from 0 to 100. For example, when you flip a coin, the probability of getting a head is 1 (it can only occur one way) divided by 2 (total possible outcomes), or 1/2. This may also be expressed as a 50% probability. P(event) = number of positive outcomes total number of outcomes Complement of an Event The complement of an event is the set of all outcomes that do not lead to the event. If you know the probability of an event, you can find the probability of its complement by subtracting the probability of the event from 1. For example, given that the probability of rolling a 5 is 1/6, the probability of its complement – rolling 1, 2, 3, 4 or 6 – is equal to 1-1/6 or 5/6. P(~event) = 1 – P(event) Dependent and Independent Events Two events are independent if the occurrence of one does not affect the probability of the other. Coin tosses and the roll of dice are examples of independent events. Two events are dependent if the outcome or occurrence of the first changes the probability of the second. For example, the odds of picking an ace from a shuffled deck of 52 playing cards are 4/52 or 1/13. If the card is not replaced, the odds of picking a second ace are changed to 3/51. The odds of picking any other particular card are also affected, because the total number of possible outcomes has changed from 52 to 51. Mutually Exclusive Events Mutually exclusive events are events that can‟t happen simultaneously, such as getting both heads and tails on the same coin toss. When two events are mutually exclusive, the probability that one or the other will occur is the sum of the probability of each event. For example, when you roll a six-sided die, the probability of getting either a 3 or a 4 is the sum of the probability of each, or 1/6 + 1/6= 2/6 or 1/3. Probability of Multiple Events To find the probability of two independent events that occur in sequence, multiply the probabilities of the events occurring separately. For example, the odds of tossing three heads in a row are ½ x ½ x ½ or 1/8. 25 Events vs. Outcomes When you roll a six-sided die, there are six equally-possible outcomes. Rolling the number four, for example, is an event that corresponds to the outcome of 4, and therefore has a 1/6 chance of occurring. You can combine outcomes to define different events with different probabilities. For example, when rolling a six-sided die, an even number is an event that corresponds to three outcomes (rolling 2, 4 or 6). Therefore the probability is 3/6 or 50%. 29. Outcomes can frequently be determined using a systematic list, a counting tree or the counting principle. In looking at the example from the probability of multiple events, what is another way that you could determine the probability of tossing three heads in a row? Probability of Two Events 30. Using a standard deck of cards, suppose you pick a card, replace it in the deck, shuffle the deck and pick a second card. What is the probability that you will pick a 6, and then another 6? 31. Using a standard deck of cards, suppose you pick a card, shuffle the deck and pick a second card without putting the first card back. What is the probability that you will pick a 6, and then another 6? 32. You are playing cards. The only cards left are a 2, 3, 3, 4, 4, 6, 8, 9, and 10. If you choose two cards at random from the stack, what is the probability that you will pick 3 and then 8? Fundamental counting principle If a first event can occur in a ways and a second event can occur in b ways (independently), then the two events can occur together in a b ways. Ex. outfits, meal possibilities 33. The Z-1 sports car comes in two different body types, convertible and hard top. The Z-1 is available in five different colors. How many choices are there for the Z-1? 26 Permutations A permutation is an arrangement of items and events in which each order of items counts as a different arrangement. Ex. combination lock, order of a line, finishing a race, books on a shelf, singing songs. Formula for Permutations n! Pm where n is the number of items and m is the number of items taken at a time. 0! is defined as 1. ( n m)! n 34. Four cyclists are racing. The first two cyclists win different prizes. How many possible permutations are there for the two prize winners? 35. How many different ways can you arrange seven people in a line? Combinations A combination is an arrangement or group in which the different order of items doesn‟t count as a different arrangement. Formula for Combinations n! C m where n is the number of items and m is the number of items taken at a time. 0! is defined as 1. (n m)!m! n 36. There are two seats left on an airplane. Five people still want to get on the flight. How many different possible pairs of passengers can be made from the five who are waiting? 27 Patterns, Algebra and Functions Questions 9-12 and related materials 28 Functions A special kind of relation in which the value of one variable depends on the value of another variable. Like a relation, it is a set of ordered pairs. However, in a function, each first value may be paired with one and only one second value. What this looks like in a table: (cannot contain repeated x values that have different y values) Are the following functions? 37. 38. 39. 40. x y x y x y x y 1 1 1 3 1 1 1 1 2 2 2 4 2 1 1 2 3 3 1 3 3 1 1 3 4 4 2 4 4 1 1 4 What this looks like in a graph: (vertical line test) According to the vertical line test, if each vertical line drawn through any point on the graph of a relation intersects the graph at no more than one point, then that relation is a function. Domain and Range Domain: In a function f(x), the possible values for x in the given situation. It is the set of values of the independent variable of a given function. Range: The possible values for y in a function. (the value of f(x)) Writing expressions from Patterns 41. There are several ways to represent algebraic relationships (written, graphically, geometrical, numerical, and as an expression). Using the pattern below, how many dots will there be in the 5th stage, in the 100th stage, in the nth stage? 29 Systems of Equations and Inequalities Would it be cheaper over the course of 4 years to pay more for a newer car and spend less for gasoline or pay less for an older car that uses a lot of gas? When would the cost of new windows in your home be paid for by the savings in heating and air conditioning costs? This type of information described by two of more related equations, called a system of equations. The point at which the graphs for these equations intersect is a solution to the system. The same applies to the intersection of the graphs of two linear inequalities, but the overlap of the graphs will be a shaded region on the graph rather than a point or points. Systems of equations can be solved in several ways, graphically, by substitution or by addition method. Typically the solution to a system of linear inequalities is show graphically. 42. In the fall, the math club and the science club each created an Internet site. You are the webmaster for both sites. It is now January and you are comparing the number of times each site is visited daily. Science club: There are currently 400 daily visits and the visits are increasing at a rate of 25 daily visits per month. Math club: There are currently 200 daily visits and the visits are increasing at a rate of 50 daily visits per month. When will the number of visits to the two sites be the same? 43. The value of you EFG stock is three times the value of you PRQ stock. If the total value of the stocks is $4500, how much is invested in each company? 44. You are planning the menu for your restaurant. Each evening you offer two different meals and have at least 240 customers. For Saturday night you plan to offer roast beef and teriyaki chicken. You expect that fewer customers will order the beef than will order the chicken. The beef costs you $5 per serving and the chicken costs you $3 per serving. You have a budget of at most $1200 for meat for Saturday night. 30 Functions What happens to the graph of a function when you take the inverse or the opposite of the function? How do the constants and coefficients relate to the graph of a function? Linear function y = mx + b m = slope, b = y intercept Quadratic function b y = ax2 + bx + c a is positive opens up, a is negative opens down, vertex = 2a parent function y = x2 Cubic function y = ax3 + bx2 + cx + d parent function y = x3 Exploration: What do you notice with the sets of graphs? y = x2 y = x2 y = x3 – 3x 2 y=x +2 x = y2 y = -(x3 – 3x) 2 y=x –2 y = -x2 y = x2 + x + 2 Write a summary statement describing how the graphs change and how that relates to the changes in the equations. 31 Systems of Linear Equations and Inequalities 45. You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium throw spotlights and at least 1 long throw spotlight. A medium throw spotlight costs $1000 and a long throw spotlight costs $3500. The minimum order for free delivery is $10,000. Write and graph a system of linear inequalities that shows how many of each type of spotlight must be ordered to get free delivery. Will an order of 4 medium throw spotlights and 1 long throw spotlight be delivered free? 46. A monthly magazine is hiring reporters to cover school and local events. In each magazine, the managing editor wants at least 4 reporters covering local news and at least 1 reporter covering school news. The budget allows for not more that 9 different reporters‟ articles to be in one magazine. Graph the region that shows the possible combinations of local and school events covered in a magazine. 47. Suppose you can work a total of no more than 20 hours per week at your two jobs. Baby-sitting pays $5 per hour and your cashier job pays $6 per hour. You need to earn at least $90 per week to cover your expenses. Write a system of linear inequalities that shows the various numbers of hours you can work at each job. Graph the result. Give two possible ways you could divide your hours between the two jobs. 48. You have 50 tickets to ride the Ferris wheel and the roller coaster. If you ride 12 times, using 3 tickets for each Ferris wheel ride and 5 tickets for each roller coaster ride, how many times did you go on each ride? 49. You spend $13 to rent 5 movies for the weekend. Since new releases rent for $3 and regular movies rent for $2, how many regular movies did you rent? How many new releases did you rent? 50. You buy six bags of wild bird food to fill the feeders in your yard. Oyster shell grit sells for $4.00 a bag and sunflower seeds sell for $4.45 a bag. If you spent $25.80, how many of each type of feed are you buying? 32 Writing and analyzing equations of a line The equation of a line can be written in the form of y = mx + b, where m is the slope and b is the y intercept. y y1 The slope is rise over run or m 2 x 2 x1 Equations of lines can also be written in standard form: ax + by = c where a, b and c are real numbers and a and b are not both zero. An equation of a line can be written from several pieces of information, a situation, a graph, a table of values, two points, or a point and the slope. Lines can be analyzed by looking at their slopes and y intercept. Two lines can be in several configurations, they can be parallel, perpendicular, intersecting, or coincident. Parallel lines will have the same slope but not the same y intercept. Perpendicular lines will have slopes that are the negative reciprocal of each other (ex. ½ and –2) Coincident lines or lines that coincide are the same line. Both the slope and the y intercept will be the same. Intersecting lines will be lines that have different slopes. 51. In 2000, the population of South Carolina was approximately 3,486,000. During the next 5 years, the population increased by approximately 37,400 people per year. Write an equation to model the population of South Carolina in terms of the number of years since 2000. Estimate the population of South Carolina in 2006. 52. Looking at the graph given below, write an equation of the line. 33 53. Given the table below, write an equation to model the total charge in dollars in terms of the number of miles driven. Miles (x) 25 50 75 100 Cost (y) 36.25 42.50 48.75 55 54. Write an equation of the line that passes through the point (6,-3) and has a slope of –2. 55. Between 1985 and 1995, the number of vacation trips in the US taken by US residents increased by about 26 million per year. In 1993, US residents went on 740 million vacation trips within the US. Write a linear equation that models the number of vacation trips in terms of the number of years since 1985. Estimate the number of vacation trips in the year 2005. 56. Write the equation of the line that passes through the points (1,6) and (3,-4). 57. While working at an archaeological dig, you find an upper leg bone (femur) that belonged to an adult human male. The bone is 43 cm long. In humans, femur length is linearly related to height. To estimate height of the person, you measure the femur and the height of two complete adult male skeletons found at the same excavation. Person 1: 40 cm femur, 162 cm height Person 2: 45 cm femur, 173 cm height Write an equation to show the relationship between height and femur length. Estimate the height of the person whose femur was found. 58. Write –4x + 3y = -6 in y intercept form. 2 59. Write y = x – 3 in standard form. 5 60. Are the pairs of lines parallel, perpendicular, coincident, or intersecting (not perpendicular). y = -3x + 2 y + 6x –8 = 0 y – 2x = 4 y = 3x + 4 y + 3x = -4 2y = 12x – 4 y=-½x+2 3y – 9x = 12 34 Geometry and Measurement Questions 13-16 and related materials 35 Unit conversion The unit cancellation method helps you convert from any unit to another unit. Ex. Change 15 ft to yards Find the relationship between the first and the second unit. (this may be in the form of one or more conversion factors). Write the known information as a ratio (over one if a single unit) Multiply by the ratio of the conversion factor(s) such that the units cancel (top to bottom or bottom to top) Multiply the numbers and cancel the units to find the solution. 1 yd = 3 ft 15 ft 1yd 15 x1 x 5 yd 1 3 ft 1x3 Try these examples 61. 25 plops is equal to how many zigs (14 zigs = 11 plops) 62. 0.057 km to yards 63. 257 cubic cm to gallons 64. 65 miles/hour to feet per second 65. 54 cubic feet to cubic yards. 36 Application of Volume, Surface Area and length of edges 66. A delivery service charges $0.20 per cubic inch for foam filler to package fragile items. Suppose a breakable parcel that was originally 6 in by 4 in by 4 in box needed to be put in a larger box. How much is the foam filler if one 4 in dimension is tripled? How much is the foam filler if both 4 in dimensions need to be tripled? (Assume the original box contained filler already and you can use that filler when you repack the box.) What is the ratio of the volume of the larger box to the volume of the original box? 67. Given a cone that has a height of 18 cm and a circular bas with a diameter of 8cm, describe a cylinder that has 3 times the volume of that cone. 68. A triangular pyramid has an equilateral triangle base and three other faces. What is the surface area of this pyramid? What is the volume of this pyramid? 69. The funnel at the right is used to put coolant in the radiator of a car. What is the maximum volume of coolant that the funnel can hold? (figure not drawn to scale) 14.5 cm 12 cm 11 cm 1.5 cm 70. Sioux tepees are cone shaped. If the diameter of a tepee is 18 ft and the height is 12 ft, how much buffalo hide is needed to cover the outside surface? 71. Denise needs to make a model of a cinder cone volcano for a school play. The cone should be 4 ft tall, the base 4 ft in diameter, and the slant height about 4.48 ft. She is going to build a frame for the cone with wire mesh. How much wire mesh will she need to buy? 37 72. The box for a videotape is 19cm tall, 10.5 cm long, and 2.5 cm wide. It is open on one of the long and narrow sides so that the tape can be put in. How much cardboard is needed to build the box? 73. You are building a model of a house. You will make the model as shown below. How much wood dowel will you need to build the frame for the model of the house? 74. A dinnerware factory packs its dinnerware in a pyramid shaped box with a square base of 12 inches by 12 inches and a height if 10 inches. How much cardboard is saved using a pyramid design over a square based prism design? How much volume is saved? 38 Distance and Midpoint Formula Distance Formula The distance between (x1,y1) and (x2,y2) is d x2 x1 2 y 2 y1 2 Midpoint Formula x1 x 2 y1 y 2 The midpoint between (x1,y1) and (x2,y2) is , 2 2 75. Find the distance between (1,4) and (-2, 3). 76. Decide whether the points (3,2), (2,0) and (-1,4) are vertices of a right triangle. 77. Find the midpoint between (-2,3) and (4,2). Use a graph to check the result. 78. You have a triangle with vertices at A(-3,-2), B(1,2), and C( 2,-2). If you were to draw a new triangle with vertices at each of the midpoints of the sides of the original triangle, how would the perimeters of the triangles compare? 79. You and your friend go hiking. You hike 3 miles north and 2 miles west. Starting from the same point, your friend hikes 4 miles east and 1 mile south. How far apart are you and your friend? If you want to meet your friend for lunch, where could you meet so that both of you hike the same distance? How far do you have to hike? 80. Draw a polygon whose vertices are A(1,1), B(5,9), C(2,8), and D(0,4). Show that the polygon is a trapezoid by showing that only two of the sides are parallel. Use the distance formula to show that the trapezoid is isosceles. 39 Mathematical Processes and Reasoning Questions 17-20 and related materials 40 Counterexamples, arguments, inductive and deductive reasoning, if…then statements (conditional statements), validity If-Then Statements: An if-then statement is just what the name says it is. It is a statement that proves if something happens then something else will happen. These kinds of if-then statements are called conditional statements, or just conditionals. Let p represent the hypothesis and q represent the conclusion, ex. if p, then q. For example, “If Chris went to the store after school, then he will buy something.” Or “If D is between C and E, then CD + DE = CE” Converse: Switching the hypothesis and the conclusion forms a converse of a conditional. Statement: If p, then q. Converse If q, then p. A statement and a converse say different things. Some converses come out to be false while the statement is true. For example, Statement: “If Chris lived in California, then he lives East of the capital of the USA” Converse: “If Chris lives East of the capital of the USA, then he lives in California” As anyone can see, the converse is false. If Chris lives east of the capital, there are a number of states he could live in, not just California. In cases such as these, where the hypothesis (p) is true and the converse (q) is false, there is a name. The names of these cases are called counterexample. Inverse: The inverse is simply the conditional with „not‟ added to it. Contrapostive: An inverse but switched around with the p and q. For example, Summary Statement: If p, then q Inverse: If not p, then not q Contrapositive: if not q, then not p Converse: If q, then p. The statement is always true with the contrapositive, but a statement is not logically equivalent to its converse or to its inverse. 41 Deductive and Inductive Reasoning In traditional Aristotelian logic, deductive reasoning is inference in which the conclusion is of no greater generality than the premises, as opposed to abductive and inductive reasoning, where the conclusion is of greater generality than the premises. Other theories of logic define deductive reasoning as inference in which the conclusion is just as certain as the premises, as opposed to inductive reasoning, where the conclusion can have less certainty than the premises. In both approaches, the conclusion of a deductive inference is necessitated by the premises: the premises can't be true while the conclusion is false. (In Aristotelian logic, the premises in inductive reasoning can also be related in this way to the conclusion.) Deductive Reasoning - A form of reasoning by which each conclusion follows from the previous one; an argument is built by conclusions that progress towards a final statement. Inductive Reasoning - A form of reasoning in which a conclusion is reached based on a pattern present in numerous observations. Examples of Deductive Reasoning Valid: All men are mortal. Socrates is a man. Therefore Socrates is mortal. The picture is above the desk. The desk is above the floor. Therefore the picture is above the floor. E comes before F in the alphabet system B and D comes before E Thus BD comes before F A man will die after a headshot A man just got shot in the head Therefore that man will die. All birds have wings. A cardinal is a bird. Therefore a cardinal has wings. Invalid: Every criminal opposes the government. Everyone in the opposition party opposes the government. Therefore everyone in the opposition party is a criminal. This is invalid because the premises fail to establish commonality between membership in the opposition party and being a criminal. This is the famous fallacy of the undistributed middle. The basic principles of logic center on the law of contradiction, which states that a statement cannot be both true and false, and the law of the excluded middle, which stresses that a statement must be either true or false. 42 Examples of Inductive Reasoning Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument support the conclusion but do not ensure it. It is used to ascribe properties or relations to types based on limited observations of particular tokens; or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is used, for example, in using specific propositions such as: This ice is cold. A billiard ball moves when struck with a cue. ...to infer general propositions such as: All ice is cold. There is no ice in the Sun. For every action, there is an equal and opposite reaction. Anything struck with a cue moves. Counterexample An example, which disproves a proposition. For example, the prime number 2 is a counterexample to the statement "All prime numbers are odd." 43 Implication in detail Clearly you can build a valid argument from true premises, and arrive at a true conclusion. You can also build a valid argument from false premises, and arrive at a false conclusion. The tricky part is that you can start with false premises, proceed via valid inference, and reach a true conclusion. For example: Premise: All fish live in the ocean Premise: Sea otters are fish Conclusion: Therefore sea otters live in the ocean There's one thing you can't do, though: start from true premises, proceed via valid deductive inference, and reach a false conclusion. We can summarize these results as a "truth table" for implication. The symbol "=>" denotes implication; "A" is the premise, "B" the conclusion. "T" and "F" represent true and false respectively. Truth Table for Implication Premise Conclusion Inference A B A => B false false true false true true true false false true true true If the premises are false and the inference valid, the conclusion can be true or false. (Lines 1 and 2.) If the premises are true and the conclusion false, the inference must be invalid. (Line 3.) If the premises are true and the inference valid, the conclusion must be true. (Line 4.) So the fact that an argument is valid doesn't necessarily mean that its conclusion holds -- it may have started from false premises. If an argument is valid, and in addition it started from true premises, then it is called a sound argument. A sound argument must arrive at a true conclusion. 44 Logic Problem Site http://pages.prodigy.net/spencejk/yearlylps.html 81. January Young People's LP#1 - New Year‟s Birthdays The five Morgan children were all born on January 1st, New Years Day, two years apart (ages 8, 10, 12, 14, and 16). This year for their birthdays, each received a different gift from their parents. From the clues, determine each child‟s name, age, and the gift he or she received for a birthday present. 1. Kristol is younger than the child who received the banjo, who is not the oldest of the five children. 2. Jason is not the child who received a Nintendo game for his or her birthday gift (who is older than Brent). 3. The girl who received the bicycle is younger than Brent, who isn‟t the child who received the telescope. 4. Teddy, who didn‟t receive the digital camera, is older than the child who received the telescope, who in turn is older than Amy. 5. It was not the oldest child who received the Nintendo game. CHILD'S NAME CHILD'S AGE GIFT RECEIVED 82. Take the following scenario: Every time a batter reaches first base, the next batter hits a double. Every time a batter hits a double, the runner on first scores. Jon reaches first base. What can you deduce about Jon? 83. Take the following scenario: When the sun shines, the grass grows. When the grass grows, it needs to be cut. The sun shines. What can you deduce about the grass? 84. Take the following scenario: Jim is a barber. Everybody who gets his hair cut by Jim gets a good haircut. Austin got a good haircut. What can you deduce about Austin? 85. All dogs are mammals, and all mammals are vertebrates. Shaggy is a dog. What can be deduced about shaggy? 86. Why is the following example of deductive reasoning faulty? Given: Khaki pants are comfortable. Comfortable pants are expensive. Adrian's pants are not khaki pants. Deduction: Adrian's pants are not expensive. 45 Logic Exercises 87. 1. If Jane has a cat, then Jane has a pet 2. Jane has a cat 3. Therefore, Jane has a pet 88. 1. If Jane has a cat, then Jane has a pet 2. Jane has a pet 3. Therefore, Jane has a cat 89. 1. If Jane has a cat, then Jane has a pet 2. It is not the case that Jane has a pet 3. Therefore, it is not the case that Jane has a cat 90. 1. If Jane has a cat, then Jane has a pet 2. It is not the case that Jane has a cat 3. Therefore, it is not the case that Jane has a pet 91. 1. If pigs fly, then hell has frozen over 2. Pigs fly 3. Therefore, hell has frozen over 92. 1. If Bush is president, then a Republican is president 2. A Republican is president 3. Therefore, Bush is president 93. 1. If E.T. phones home, then blue is Joe's favorite color 2. It is not the case that blue is Joe's favorite color 3. Therefore, it is not the case that E.T phones home 94. 1. It is not the case that Yoda is green 2. If Darth Vader is Luke's Dad, then Yoda is green 3. Therefore, it is not the case that Darth Vader is Luke's dad 95. 1. Dan plays the cello 2. If Mary plays the harp, then Owen plays the clarinet 3. Therefore, it is not the case that Mary plays the harp 96. 1. All smurfs are snorks 2. All ewoks are snorks 3. Therefore, All smurfs are ewoks 46 97. 1. Kate is a lawyer 2. Therefore, Kate is a lawyer 98. 1. If it is morally permissible to kill an 8-month old fetus, then it is morally permissible to kill a newborn infant 2. It is not the case that it is morally permissible to kill a newborn infant 3. Therefore, it is not the case that it is morally permissible to kill an 8-month old fetus 99. 1. If Rufus is a human being, then Rufus has a right to life 2. It is not the case that Rufus is a human being 3. Therefore, it is not the case that Rufus has a right to life 100. 1. All anarchists are socialists 2. All socialists are totalitarians 3. Therefore, all anarchists are totalitarians 101. 1. No cat is a biped 2. All kangaroos are bipeds 3. Therefore, No cat is a kangaroo 102. 1. If there is order in the universe, then God exists 2. There is order in the universe 3. Therefore, God exists 103. 1. Amy joins the Army, or Mary joins the Marines 2. It is not the case that Mary joins the Marines 3. Therefore, Amy joins the Army (Note: the word 'OR' is a logical term much like 'if…then', 'therefore' and 'it is not the case that…' Like these other terms, 'OR' is part of the structure or form of the argument, rather than the content.) 104. 1. Ariel joins the Air Force or Nancy joins the Navy 2. Nancy joins the Navy 3. Therefore, Ariel joins the Air Force 47 48 Time Trial #1 1. The table shows the number of medals won by 24 countries at the 1998 Winter Olympics. What was the least number of medals won by a country in the top quartile? A. 1 medal B. 2 medals C. 12 medals D. 13 medals 2. A businesswoman calculates that the median cost of her five business trips last month was $600. Which of these statements is correct? A. She spent a total of $3,000 on business trips last month. B. She spent $600 on most of the business trips last month. C. She spent $600 or more on at least half of the business trips last month. D. She spent $600 more on her most expensive business trip than she did on her least expensive trip. 49 3. The growing season is defined as the average number of days between the last frost in the spring and the first frost of the fall. The table below shows the average growing season for nine major U.S. cities. If the circled number 156 is changed to 256, how would the mean, median, and mode of this data change? A. increase mean, median, and mode B. decrease mean, median, and mode C. increase mean, median, eliminate mode D. increase mean and mode, decrease median 4. How many ways can you arrange 7 different books, so that a specific book is on the third place? A. 5040 different ways B. 720 different ways C. 42 different ways D. 7 different ways 5. (x + 5) (7 + x) = (x + 5 ) (7) + (x + 5 ) (x) is an example of which property? A. associative property of addition B. distributive property C. commutative property of multiplication D. associative property of multiplication 50 6. Tim Molloha owns and operates the Dancing Feet dance studio in downtown Saltsburg. There he trains the best and most promising young men and women to dance. His specialty is Irish Tap dancing, although he has been know to sprinkle in a class or two of ballet from time to time. He charges $15.75 for each person for a lesson and all of his classes are full with 9 people in a class. He runs 5 classes per day, 5 days a week, 45 weeks out of the year. Which expression below would you use to calculate the amount of money Tim earns in 2 years? 45weeks 5days 5classes $15.75 9 people A. 2 years 1year week day person class year week day $15 .75 9 people B. 2 years 45 weeks 5days 5classes class class 5days $15.75 9 people C. 2 years 45weeks 5classes class 45 weeks 5days 5classes $15 .75 D. 2 years 1year week day class 7. Kathy uses 3 cups of sugar and 8 cups of flour to make chocolate chip cookies. She needs to make a mix of 44 cups of dry mix. How much flour does she need? A. 16 ½ cups B. 32 cups C. 41 cups D. 117 cups 8. An athletic director asked 4 groups of students, “What is the most that you would pay for an athletic event?” The responses are shown below. Which group's answers showed the least variability? A. Group 1 B. Group 2 C. Group 3 D. Group 4 51 9. In the United States, 43% of people wear a seat belt while driving. If two people are chosen at random, what is the probability that both of them wear a seat belt? A. 86% B. 18% C. 57% D. none of the above Extended Response #1 10. a. Seven miles of highway between Tucson and Phoenix are being repaved and expanded from 2 lanes to 3 lanes. You are the project supervisor and are preparing to buy the paint needed for striping the new road. Assume that all of the stripes will be painted the same color. Each gallon of the reflective paint will paint 400 yards of the lane separator stripe or 150 yards of the edge stripe. How much paint will you need to order to complete the project? b. Assuming that you will need to order paint for other projects as well, write a formula to help you figure out how much paint to order no matter the length of the road or the number of lanes. 52 Time Trial #2 1. A concert promoter keeps track of advance ticket sales to estimate attendance at a performance and determine how much help is needed in planning the event. Which equation best represents the data given in the table? Advance Ticket Sales (A) Actual Attendance (n) 5628 13,581 7043 15,902 8912 19,873 9117 21,683 9741 22,705 A. n = 0.5A + 10,000 B. n = 2A2 C. n = 2.275A – 50 D. n = 0.75A 2. Suppose that one music subscription service, Dozster, charges $11 per month plus $0.85 per song and a second service, Melody, charges $8 per month plus $1 per song. If you were to download 50 songs per month, which statement is true? A. using the Dozster service would be less expensive B. using the Melody service would be less expensive C. the cost for using the services would be the same D. you cannot determine which company would be less expensive from the information given 3. The figure below shows a right triangle ABC with right angle at A and an altitude drawn from vertex A to the hypotenuse BC. What is the length of the hypotenuse BC? B 25 D y 65 x A C 156 A. 60 B. 85 C. 144 D. 169 53 4. Assume that when a dart is thrown and hits the board, the dart is equally likely to hit any point on the board. Given that the diameter of the inner circle is 2, the radius of the middle circle is 5, and the diameter of the outer circle is 18, what is the probability that the dart will land between the inner and outer circle in the shaded area? A. 50% B. 30% C. 25% D. 70% 5. Solve for x. 3 x 1 x 4 A. 3, 4 B. –3, -4 C. –4, 3 D. –3, 4 6. Without grouping symbols, the expression 2 33 4 has a value of 58. Which expression given below contains grouping symbols such that the value would be 220? A. 2(3 3 4) B. (2 3) 3 4 C. (2 3 3 ) 4 D. (2 3) (3 4) 7. A plumber charges a basic service fee plus a labor charge for each hour of service. A 2-hour job costs $120 and a 4-hour job costs $180. What is the plumber‟s basic service fee? A. $60 B. $80 C. $100 D. $120 54 8. Which of the following represents a function? I. II. III. Input Output Input Output Input Output 1 4 1 3 1 3 2 4 2 3 1 -3 3 6 3 4 2 4 4 6 4 4 3 5 A. All B. I and II C. I and III D. II and III E. None 9. Which choice shows the given equation so that y is a function of x? 2y – 3(y – 2x) = 8(x – 1) + 3 5 y y 5 A. x B. x 2 10 C. y 14x 5 D. y 2x 5 55 10. Given the map of the orchard shown below, what is the area of the region where the oak trees are planted? maple elm oak sycamore pine A. 4 square units B. 64 square units C. 8 square units D. 32 square units 56 Extended Response Task #2 A developer is building houses with backyards that are rectangular in shape and twice as long as they are wide. Each backyard will be landscaped with a grass area that is one-fourth the length of the yard but spans the entire width. The remainder of the yard will be covered with pea gravel. (see figure) Grass area Gravel area 1. Write a general equation for: a. The area of the entire yard b. The area of the grass area c. The area of the gravel area 2. Calculate the ratio of the gravel area to the grass area. 3. If your entire yard were 5000 square feet, how many square feet of sod would need to be purchased to landscape the grass area? 57 58 Simplifying Radical Expressions 1. Simplify the expression 50 . Prime factorization of 50 = 2 5 5 50 = 2 5 5 = 5 2 2. Simplify the expressions. 3 20 32 a. b. c. 4 4 50 3 225 32 4 4 50 3 2 5 22222 22 4 255 3 5 4 2 4 = 2 2 5 2 5 3. Body Surface Area Physicians can approximate the Body Surface Area of an adult (in sq. meters) using an index called BSA where H is height in cm and W is weight in kg. HW Body Surface Area (BSA) = 3000 Find the BSA of a person who is 160 cm tall and weighs 50 kg. 160 50 BSA = 3000 8000 BSA = 3000 8 BSA = 3 8 BSA = 3 222 3 BSA = 3 3 2 6 BSA = 3 59 Solving Quadratic Functions 4. Algebraically Some simple quadratic equations can be solved algebraically. 1 2 x 8 2 2 1 2 2 x 8 1 2 1 x 2 16 x 2 16 x 4 5. Using a Graph The solutions, or roots, of ax2 + bx + c = 0 are the x-intercepts of the graph of the parabola. x2 x 2 x2 x 2 0 x y -2 4 -1 0 0 -2 1 -2 2 0 60 6. Factoring Solve x 2 3x 4 0 . (x +4) (x-1) = 0 Either (x + 4) or (x – 1) or both must equal zero to make the equation true. If x + 4 = 0 then x = -4, if x-1 = 0 then x = 1. Using Quadratic Equation 7. Solve 2x2 - 4x – 3 = 0, and round your answer to two decimal places, if necessary. a=2 b = -4 c = -3 (4) (4) 2 4(2)(3) x 2(2) 4 16 24 x 4 4 40 x 4 4 6.32 x 4 4 6.32 4 6.32 x OR x 4 4 x = 2.58 OR –0.58 61 8. x(x-2) = 4 Hint: Make sure to write the equation in the form of quadratic = zero. x2 - 2x = 4 x2 - 2x – 4 = 0 a=1 b = -2 c = -4 (2) (2) 2 4(1)(4) x 2(1) 2 4 16 x 2 2 20 x 2 22 5 x 2 22 5 22 5 x OR x 2 2 x 1 5 OR x 1 5 62 Applications of Quadratic Equations Vertical Motion Models Object is dropped h = -16t2 + s Object is thrown h = -16t2 + vt + s h = height (feet) t = time in motion (seconds) s = initial height (feet) v = initial velocity (feet per second) In these models, the coefficient of t2 is one half the acceleration due to gravity. On the surface of the Earth, this acceleration is approximately 32 feet per second per second. Use the vertical motion model to find how long it will take for the object to reach the ground. 9. You drop keys from a window 30 feet above ground to your friend below. Your friend does not catch them. h = -16t2 + s s = 30 feet h = when keys hit ground = 0 feet 0 = -16t2 + 30 16t2 = 30 30 t2 = 16 30 t 4 63 10. A lacrosse player throws a ball upward from her playing stick with an initial height of 7 feet, at an initial speed of 90 feet per second. h = -16t2 + vt + s Solving for t – the amount of time it takes for the ball to hit the ground s = 7 feet h = when ball hits the ground = 0 feet v = 90 ft/sec 0 = -16t2 + 90t + 7 a = -16 b = 90 c=7 use decimal approximation and quadratic equation (90) (90) 2 4(16)(7) x 2(16) 90 8100 448 x 32 90 8548 x 32 90 92.45 x 32 90 92.45 90 92.45 x OR x 32 32 x = -0.08 s or 5.70 s x = 5.7 seconds since only the positive solution applies 64 Prime Factorization 11. Find the prime factorization of 132. 12. Find the prime factorization of 72. 2 2 3 11 23 32 13. Find the prime factorization of 105. 357 Greatest Common Factor 14. Find the GCF of 18 and 30. GCF of 18 and 30 is 6. 15. Find the GCF of 96 and 144. GCF of 96 and 144 is 48. Least Common Multiple 16. Find the LCM of 8, 9, and 12. The LCM of 8, 9 and 12 is 72. 17. Find the LCM of 18 and 30. The LCM of 18 and 30 is 90. Applications of LCM and GCF 18. After moving forward 108 inches the same points P and Q will be touching the ground, LCM 19. When the tires have moved forward 234 inches, Karen will have done 13 revolutions and Maria will have done only 9, LCM 20. teams of 6, GCF 21. 15 packages, GCF 22. 120 of each item needed, 12 pkg hot dogs, 15 pkg buns, 5 pkg ice cream, LCM 23. 6 plants in a group, GCF 24. 90 days, LCM 25. 250,755 days, LCM 65 Standard Deviation Problems 26. A normally distributed set of data has a mean of 20 and a standard deviation of 3. Determine the probability that a randomly selected x-value is between 14 and 26. Using the percents that describe the ranges of values in a normal distribution, 68% of the data for the curve will fall within 1 standard deviation of the mean in the range from 17 (20 – 3) to 23 (20 + 3). 97% of the values will fall within two standard deviations of the mean in the range from 14 (20-3-3) to 26 (20+3+3). The probability of selecting a value between 14 and 26 would be 97% or 0.97. 27. A normally distributed set of data has a mean of 60 and a standard deviation of 10. What percent of data fall between 50 and 70? Using the percents that describe the ranges of values in a normal distribution, 68% of the data for the curve will fall within 1 standard deviation of the mean in the range from 50 (60 – 10) to 70 (60 + 10). 68% of the values will fall in this range. 28. In a research project, it was found that the heights of coconut trees are normally distributed with a mean of 60 feet and a standard deviation of 5 feet. Determine the probability that five randomly selected coconut trees are all between the heights of 50 and 70 feet. Using the percents that describe the ranges of values in a normal distribution, 68% of the data for the curve will fall within 1 standard deviation of the mean in the range from 55 (60 – 5) to 65 (60 + 5). 97% of the values will fall within two standard deviations of the mean in the range from 50 (60-5-5) to 70 (60+5+5). The probability of selecting a value between 50 and 70 would be 97% or 0.97. Probability Problems Events vs. Outcomes 29. What is the probability of tossing three heads in a row using a fair coin? Systematic List There are 8 possible outcomes to flipping a coin three times. HHH HHT HTH THH HTT THT TTH TTT Three heads in a row is one of these 8 outcomes for a probability of 1/8. P(H,H,H) = ½ x ½ x ½ = 1/8 66 Probability of Two Events 30. Using a standard deck of cards, suppose you pick a card, replace it in the deck, shuffle the deck and pick a second card. What is the probability that you will pick a 6, and then another 6? P (6, 6) = 4/52 x 4/52 = 16/2704 = 0.59% 31. Using a standard deck of cards, suppose you pick a card, shuffle the deck and pick a second card without putting the first card back. What is the probability that you will pick a 6, and then another 6? P (6, 6) = 4/52 x 3/51 = 12/2652 = 0.45% 32. You are playing cards. The only cards left are a 2, 3, 3, 4, 4, 6, 8, 9, and 10. If you choose two cards at random from the stack, what is the probability that you will pick 3 and then 8? P (3, 8) = 2/9 x 1/8 = 2/72 = 2.8% Counting Principle 33. The Z-1 sports car comes in two different body types, convertible and hard top. The Z-1 is available in five different colors. How many choices are there for the Z-1? Systematic List Each of the five colors can be made in either a convertible or hard top. There are 10 choices of color and body style combinations. Color 1 with convertible Color 1 with hard top Color 2 with convertible Color 2 with hard top Color 3 with convertible Color 3 with hard top Color 4 with convertible Color 4 with hard top Color 5 with convertible Color 5 with hard top 67 Tree Diagram Convertible Color 1 Hard Top Convertible Color 2 Hard Top Convertible Color 3 Hard Top Convertible Color 4 Hard Top Convertible Color 5 Hard Top Counting Principle Choices in set 1 x choices in set 2 = 5 x 2 = 10 possible cars combinations 68 Permutations 34. Four cyclists are racing. The first two cyclists win different prizes. How many possible permutations are there for the two prize winners? Systematic List Cyclist A, B, C, and D are the four cyclists. Using a systematic list. There are … 1st place 2nd place A B A C 12 possible placings of the 4 cyclists. A D B A B C B D C A C B C D D A D B D C Using a formula Use the permutation formula since order matters. (Cyclist A in 1st place and cyclist B in second place is not the same as Cyclist B in 1st place and cyclist A in second place) n! Pm ( n m)! n 4! 4 P2 (4 2)! 4! 4 P2 2! 4 3 2 1 4 P2 = 12 possible placings 2 1 69 35. How many different ways can you arrange seven people in a line? Since there would be 7 of these diagrams to complete all of the arrangements, the total number of line orders is 7 times the number of possibilities in one tree diagram or 5040 different line orders. This problem could be solved using a tree diagram but it would be very large. Using a Formula – since order matters use the permutation formula. N is 7 and m is 7 in this problem since all of the people in the set are being placed in the line. n! n Pm (n m)! 7! 7 P7 (7 7)! 7! 4 P2 0! = 1 0! 7 6 5 4 3 2 1 4 P2 = 5040 possible line orders 1 70 Combinations 36. There are two seats left on an airplane. Five people still want to get on the flight. How many different possible pairs of passengers can be made from the five who are waiting? Systematic List Person A, B, C, D, and E will be the people still wanting to get on the plan. Order does not matter since A,B is the same as B,A, the same two people get on the plane. A, B A, C Systematic list starting with A combinations, B combinations etc A, D without duplicates A, E B, C B, D B, E C, D C, E D, E 10 combinations of passengers possible if 2 of the 5 passengers can get on the plane. Using a formula Use the choose formula since order doesn‟t matter. N = 5 and m = 2 since 2 of the original 5 people are being chosen. n! n C m (n m)!m! 5! 5 C 2 (5 2)!2! 5! 5 C 2 (3)!2! 5 4 3 2 1 5 C 2 = 10 combinations (3 2 1)(2 1) Functions 37. Yes 38. Yes 39. Yes 40. No 71 Writing Expressions from Patterns 41. The 5th stage could be drawn or figured out from a pattern in a t-chart. There would be 17 dots in the 5th stage of the pattern. Stage Number of Dots 1 1 2 5 3 9 4 13 5 17 The 100th stage would be very difficult to build or draw so a generalization must be made. For each of the stages, four dots is added to the outside of the pattern. The number of dots in the 100th stage could be solved from the expression 1 + 4(n-1) where 1 is the number of dots in the first stage and n-1 times 4 would be the number of dots added to get to the nth stage 1 + 4(100-1) 397 dots for the 100th stage Make sure to verify that your written expression works for a known stage or two. 1 + 4 (2-1) for the 2nd stage 5 dots in the 2nd stage 1 + 4 (3-1) for the 3rd stage 9 dots in the 3rd stage To figure out a generalized equation/expression, the graph can be drawn and slope intercept form used. This can also be accomplished by taking the pattern back to a zero stage. Stage Number of Dots 0 -4 1 1 2 5 3 9 4 13 5 17 Intercept is –3 Slope is change in number of dots over change in stage = 4/1 y = 4x – 3 Where x is the stage and y is the number of dots. 72 Systems of Equations and Inequalities 42. In the fall, the math club and the science club each created an Internet site. You are the webmaster for both sites. It is now January and you are comparing the number of times each site is visited daily. Science club: There are currently 400 daily visits and the visits are increasing at a rate of 25 daily visits per month. Math club: There are currently 200 daily visits and the visits are increasing at a rate of 50 daily visits per month. When will the number of visits to the two sites be the same? t = total visits to website m = months Number of visits to the science club Number of visits to the math club t = 400 + 25m t = 200 + 50m When will number of visits be the same? When t is the same. Therefore, the visits will be the same when 400 + 25m = 200 + 50m 200 = 25m 8=m After 8 months the number of visits will be the same. (in October) 43. The value of your EFG stock is three times the value of your PRQ stock. If the total value of the stocks is $4500, how much is invested in each company? e = value of EFG stock p = value of PRQ stock Value of EFG stock three times the value of PRQ stock e = 3p Total value of the stocks $4500 e + p = 4500 Total invested in each company, since e = 3p, the second equation can be changed to 3p + p = 4500 4p = 4500 p = $1125 You hold $1125 in PRQ stock and 3 times that $3375 in EFG stock. 73 44. You are planning the menu for your restaurant. Each evening you offer two different meals and have at least 240 customers. For Saturday night you plan to offer roast beef and teriyaki chicken. You expect that fewer customers will order the beef than will order the chicken. The beef costs you $5 per serving and the chicken costs you $3 per serving. You have a budget of at most $1200 for meat for Saturday night. b = number of beef meals (x axis) c = number of chicken meals (y axis) two different meals and at least 240 customers c + b 240 Slope intercept form: c 240 – b Solid line – shade above the line Fewer customers will order the beef than will order the chicken c>b Dotted line shade above the line The beef costs $5 per serving and chicken is $3 per serving with budget of at most $1200 5b + 3c 1200 Slope intercept form: 3c 1200 – 5b 5 c 400 – b 3 Solid line shade below the line 400 The chef should order in the range of 240 to 400 chicken dinners and 240 approx 170 to 0 beef dinners. Cost for dinners purchased should not exceed $1200. Number of chicken Number 240 of beef 74 Systems of Linear Equations and Inequalities 45. You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium throw spotlights and at least 1 long throw spotlight. A medium throw spotlight costs $1000 and a long throw spotlight costs $3500. The minimum order for free delivery is $10,000. Write and graph a system of linear inequalities that shows how many of each type of spotlight must be ordered to get free delivery. Let m = the number of medium throw spotlights and let l = the number of long throw spotlights Number of medium throw spotlights m3 l 1 1000m 3500l 10,000 To graph the last inequality, write in slope intercept form (in terms of l) 1000m 10,000 3500l 1000 10000 3500 m l 1000 1000 1000 m 10 3.5l Number of long throw spotlights Will an order of 4 medium throw spotlights and 1 long throw spotlight be delivered free? 1000(4)+3500(1) = 7500. The amount of the order $7500 would not qualify for free delivery. The point (1,4) does not fall in the shaded portion of the graph. 46. A monthly magazine is hiring reporters to cover school and local events. In each magazine, the managing editor wants at least 4 reporters covering local news and at least 1 reporter covering school news. The budget allows for not more that 9 different reporters‟ articles to be in one magazine. Graph the region that shows the possible combinations of local and school events covered in a magazine. Let l = the number of reporters covering local news and let s = the number of reporters covering school news l4 s 1 ls9 Local Reporters To graph the last inequality, write in slope intercept form (in terms of s) l 9s School Reporters 75 47. Suppose you can work a total of no more than 20 hours per week at your two jobs. Baby-sitting pays $5 per hour and your cashier job pays $6 per hour. You need to earn at least $90 per week to cover your expenses. Write a system of linear inequalities that shows the various numbers of hours you can work at each job. Graph the result. Let b = the number of hours of babysitting and let c = the numer of hours of cashiering b c 20 5b 6c 90 Give two possible ways you could divide your hours between the two jobs. There are many ways to earn the $90 needed to cover the week’s expenses. If you were to babysit for all 20 hours you would be able to earn $100. If you were to cashier all 20 hours you could earn $120. The more you cashier the closer to $120 you would earn. 48. You have 50 tickets to ride the Ferris wheel and the roller coaster. If you ride 12 times, using 3 tickets for each Ferris wheel ride and 5 tickets for each roller coaster ride, how many times did you go on each ride? Let f = the number of ferris wheel rides and let r = the number of roller coaster rides. f r 12 3 f 3r 36 r7 3 f 5r 50 3 f 5r 50 f 7 12 f 5 2r 14 r7 Solving this system of equations you would have ridden on the ferris wheel 5 times and the roller coaster 7 times. 49. You spend $13 to rent 5 movies for the weekend. Since new releases rent for $3 and regular movies rent for $2, how many regular movies did you rent? How many new releases did you rent? Let n = the number of new releases and r = the number of regular movies. 3n 2r 13 3n 2r 13 r2 nr 5 3n 3r 15 n25 n3 r 2 r2 Solving this system of equations you would rented 2 regular movies and 3 new releases. 76 50. You buy six bags of wild bird food to fill the feeders in your yard. Oyster shell grit sells for $4.00 a bag and sunflower seeds sell for $4.45 a bag. If you spent $25.80, how many of each type of feed are you buying? Let o = the number of bags of oyster shell bird food and s = the number of bags of sunflower seeds. os 6 4o 4s 24 s4 4o 4.45s 25.80 4o 4.45s 25.80 o46 o2 0.45s 1.8 s4 You would have bought 4 bags of sunflower seeds and 2 bags of oyster shell feed. Writing and analyzing equations of a line 51. In 2000, the population of South Carolina was approximately 3,486,000. During the next 5 years, the population increased by approximately 37,400 people per year. Write an equation to model the population of South Carolina in terms of the number of years since 2000. Estimate the population of South Carolina in 2006. The slope in this situation is the rate of change or 37,400 people per year. The y-intercept or the starting point is the population in 2000, 3,486,000 people. The equation of the line would be: y mx b y 37400 x 3486000 Where y is the population of South Carolina and x is the number of years since 2000. The equation of the line can be used to estimate the population of South Carolina in 2006 by substituting 6 in for x (6 years after 2000). y 37400 (6) 3486000 y 3,710 ,400 people 77 52. Looking at the graph given below, write an equation of the line. From the graph the y intercept is -4 and the slope is positive 1 (up one over one). The equation of the line is: y mx b y x4 53. Given the table below, write an equation to model the total charge in dollars in terms of the number of miles driven. Miles (x) 25 50 75 100 Cost (y) 36.25 42.50 48.75 55 Given the table below, we have neither the slope nor the y intercept given. We can figure out the slope and then use the slope and an ordered pair from the table in the equation to find the y intercept. Using the points (25,36.25) as (x1,y1) Using the slope (0.25) and one of the ordered and (50,42.50) as (x2,y2) pairs (25, 36.25) as (x,y) y y1 m 2 y mx b x 2 x1 42.5 36.25 36.25 (0.25)(25) b m 50 25 36.25 6.25 b 6.25 b 36.25 6.25 m 25 b 30 m 0.25 The equation of the line would be: y 0.25x 30 . 78 54. Write an equation of the line that passes through the point (6,-3) and has a slope of –2. Given the information from the problem, we are given the slope and can solve for the y intercept using substitution. Using the slope (-2) and the ordered pair (6, -3) as (x,y) y mx b 3 ( 2)(6) b 3 12 b b 3 12 b9 The equation of the line would be: y 2 x 9 . 55. Between 1985 and 1995, the number of vacation trips in the US taken by US residents increased by about 26 million per year. In 1993, US residents went on 740 million vacation trips within the US. Write a linear equation that models the number of vacation trips in terms of the number of years since 1985. Estimate the number of vacation trips in the year 2005. In this situation, the slope is given by 26 million vacations per year. A point on the line is the data from the year 1993, since 1993 is 8 years after 1985 the ordered pair would be (8, 740 million). Using this information and substitution, we can find the y-intercept and write the equation of the line. y mx b y mx b The equation of the line is as follows 740mill (26mill)(8) b y 26 x 532 with y being the number of vacations 740mill 208mill b and x being the years since 1985. To figure out the number of vacation trips b 740mill 208mill for 2005, the value for x will be 10 (10 b 532million years since 1985) y 26 x 532 y 26(10) 532 y 26(10) 532 y 792million 79 56. Write the equation of the line that passes through the points (1,6) and (3,-4). From the ordered pairs, the slope can be calculated. Once the slope is calculated, one of the ordered pairs and the slope can be used to find the y-intercept. Using the points (1,6) as (x1,y1) Using the slope (-5) and one of the ordered and (3,-4) as (x2,y2) pairs (1, 6) as (x,y) y y1 m 2 y mx b x 2 x1 6 5(1) b 6 4 m 6 5 b 1 3 10 b 65 m b 11 2 m 5 y 5 x 11 57. While working at an archaeological dig, you find an upper leg bone (femur) that belonged to an adult human male. The bone is 43 cm long. In humans, femur length is linearly related to height. To estimate height of the person, you measure the femur and the height of two complete adult male skeletons found at the same excavation. Person 1: 40 cm femur, 162 cm height Person 2: 45 cm femur, 173 cm height Write an equation to show the relationship between height and femur length. Estimate the height of the person whose femur was found. Given two ordered pairs (40, 162) and (45, 173), you could calculate the slope and then use one of the ordered pairs and substitution to calculate the y-intercept. From this point, the equation of the line can be written and the relationship used to estimate the height of the person whose femur was 43 cm long. Using the points (40,162) as (x1,y1) Using the slope (2.2) and one Solving to estimate the and (45,173) as (x2,y2) of the ordered pairs (40, 162) height of the person as (x,y) with a femur that is 43 cm long. Y is the y 2 y1 y mx b m height and x is the x 2 x1 162 2.2(40) b femur length in the 173 162 162 88 b equation, so by m substitution: 45 40 b 162 88 y 2.2 x 74 11 b 74 m 2.2 y 2.2(43) 74 5 y 168.6cm y 2.2 x 74 80 58. Write –4x + 3y = -6 in y intercept form. To write the equation in slope intercept for we must solve for y, or put the equation in terms of x. 4 x 3 y 6 3 y 6 4 x 4 y 2 x 3 4 y x2 3 Where the slope is 4/3 and the y intercept is -2. 2 59. Write y = x – 3 in standard form. 5 Standard form of a linear equation is in the form ax + by = c. To rewrite the equation in this way, solve so that the variables are on the left and the constant is on the right side of the equation. The constant should be a whole number as should the coefficients of x and y. 2 y x3 5 2 y x 3 5 2 5 y x 3 5 5 y 2 x 15 2 x 5 y 15 60. Are the pairs of lines parallel, perpendicular, coincident, or intersecting (not perpendicular). To determine if the lines are parallel, perpendicular, coincident, or intersecting (not perpendicular) we analyze their slopes and y intercepts. In order to analyze and compare the equations, it is helpful to write all of the equations in slope intercept form. y = -3x + 2 y + 3x = -4 slope = -3 y = -3x – 4 y-int=2 slope = -3 y-int = -4 Since the slopes of the lines are the same but the y-intercept is different, the lines are parallel. y + 6x –8 = 0 2y = 12x – 4 y = -6x +8 y = 6x – 2 slope = -6 slope = 6 y-int = 8 y-int = -2 Since the slopes and the y intercepts of the lines are different, they will intersect. 81 y – 2x = 4 y=-½x+2 y = 2x + 4 slope = - ½ slope = 2 y-int = 2 y-int = 4 Since the slopes are the negative reciprocal of each other, the lines will meet at a right angle (perpendicular). y = 3x + 4 3y – 9x = 12 slope = 3 3y = 9x + 12 y-int = 4 y = 3x + 4 slope = 3 y-int = 4 Since the equations of the lines are the same (same slope and same y-int) the lines coincide (are the same line). Unit conversion 61. 25 plops is equal to how many zigs (14 zigs = 11 plops) 25 plops 14 zigs 25 x14 x 31 .82 zigs 1 11 plops 1x11 62. 0.057 km to yards km miles yd 1 mile = 1.609 km 1 mile = 1760 yd 0.057 km 1mile 1760 yards 0.057 x1x1760 x x 62 .35 yards 1 1.609 plops 1mile 1x1.609 x1 63. 257 cubic cm to gallons cubic cm mL L gallons 1 cubic cm = 1 mL 1000 mL = 1 L 1 L = 0.264 gal 257 cm 3 1mL 1L 0.264 gal 257 x0.264 x 3 x x 0.07 gal 1 1cm 1000 mL 1L 1000 64. 65 miles/hour to feet per second miles ft and hours min sec 1 mile = 5280 ft 1 hour = 60 minutes 1 min = 60 sec 65miles 5280 ft 1hour 1 min 65x5280 x x x 95.3. ft / sec 1hour 1mile 60 min 60 sec 60 x60 65. 54 cubic feet to cubic yards. Cubic feet cubic yards 27 cubic feet = 1 cubic yard 54 ft3 1yd 3 54 x 3 2 yd 3 1 27 ft 27 82 Application of Volume, Surface Area and length of edges 66. A delivery service charges $0.20 per cubic inch for foam filler to package fragile items. Suppose a breakable parcel that was originally 6 in by 4 in by 4 in box needed to be put in a larger box. How much is the foam filler if one 4 in dimension is tripled? How much is the foam filler if both 4 in dimensions need to be tripled? (Assume the original box contained filler already and you can use that filler when you repack the box.) What is the ratio of the volume of the larger box to the volume of the original box? The original box with dimensions of 6 x 4 x 4 has a volume of 96 cubic inches. The box with one 4 in dimension tripled has dimensions of 6 x 12 x 4 and a volume of 288 cubic inches. To fill this box with filler you will need 192 more cubic inches of filler by subtracting the volume of this box from the volume of the original box. 288 in3 – 96 in3 = 192 in3 Since the filler costs $0.20 per cubic inch, the cost for the 192 in3 would be $38.40. 192 in3 x $0.20/ in3 = $38.40 The ratio of the volume of this box to the volume of the original box is 3:1. (288:96) The box with two 4 in dimensions tripled has dimensions of 6 x 12 x 12 and a volume of 864 cubic inches. To fill this box with filler you will need 768 more cubic inches of filler by subtracting the volume of this box from the volume of the original box. 864 in3 – 96 in3 = 768 in3 Since the filler costs $0.20 per cubic inch, the cost for the 768 in3 would be $153.60. 768 in3 x $0.20/ in3 = $153.60 The ratio of the volume of this box to the volume of the original box is 9:1. (864:96) 67. Given a cone that has a height of 18 cm and a circular base with a diameter of 8cm, describe a cylinder that has 3 times the volume of that cone. The volume of a cone is one third times the volume of a cylinder with the same height and radius. 1 Vcone r 2 h Vcylinder r 2 h 3 Because of this relationship between the volume of a cylinder and a cone, the dimensions of a cylinder with 3 times the volume of the cone described in the problem would be a cylinder with a the same height of 18cm and same diameter of 8cm (or radius of 4cm). 68. A triangular pyramid has an equilateral triangle base and three other faces. What is the surface area of this pyramid? What is the volume of this pyramid? The surface area of the triangular pyramid is the area of all of the four triangular faces added together. The volume of the pyramid is the area of one of the triangular faces times 1/3 times the height of the pyramid. If the length of the side of the equilateral triangle is x, we can write a formula for the surface area and volume of the triangular pyramid. 83 The surface area can be found by finding the area of each face and multiplying by 4. The area of a face in terms of x can be calculated by using Pythagoreans theorem. Then the surface area of the triangular pyramid can also be written in terms of x. Height of the Triangle a2 b2 c2 Area of a Triangular Face x 2 A bh x x 3 b x 2 2 x 3 2 2 A x 2 x2 x b2 x2 4 x2 3 2 A x2 2 b2 x2 4 2 Surface Area of the Triangular 4x x2 b2 Pyramid 4 4 x2 3 3x 2 A 4 b2 2 4 A 2 3x 2 x 3 b 2 The volume of the triangular pyramid is the area of the base time the height. The height of the triangular pyramid can also be calculated using Pythagorean’s theorem. The height of the pyramid can be determined by looking at the 30-60-90 triangle whose vertices are at the center of the base triangle, the top of the pyramid and the midpoint of one of the base edges. The hypotenuse of this triangle is the same measurement of the height of x 3 the triangle and the measurement between the midpoint of the base edge and the center of the base 2 x 3 triangle is half that height , in which can we can apply Pythagorean’s theorem to figure out the height of 4 the pyramid in terms of x and then calculate the volume of the pyramid in terms of x. Height of the Pyramid a2 b2 c2 2 2 x 3 x 3 Volume of the Pyramid b2 4 2 V = Abh 3x 3x 2 3x 2 V 2 3x 2 b2 4 16 4 3 3x 3 3x 2 3x 2 V b2 2 4 16 2 12 x 3x 2 b 2 16 16 2 9x b2 16 3x b 84 4 69. The funnel at the right is used to put coolant in the radiator of a car. What is the maximum volume of coolant that the funnel can hold? (figure not drawn to scale) 14.5 cm 12 cm 11 cm 1.5 cm The volume of the funnel would be the volume of the cone that makes up the top part of the funnel plus the volume of the cylinder that makes up the bottom part of the funnel. Volume of the cone Volume of the cylinder Volume of the funnel Vcone 1 / 3r 2 h Vcylinder r 2 h Vfunnel Vcone Vcylinder Vcone 1 / 3 (7.25) 2 12 Vcylinder (. 75 ) 2 11 Vfunnel 210.25 6.1875 Vcone 210.25cm 3 Vcylinder 6.1875 cm 3 Vfunnel 216.4375 Vfunnel 680cm 3 70. Sioux tepees are cone shaped. If the diameter of a tepee is 18 ft and the height is 12 ft, how much buffalo hide is needed to cover the outside surface? Lateral Surface Area of a Cone where l = slant height = 12 ft, r = 9 ft SAcone rl SAcone (9)(12 ) SAcone 108 ft 2 SAcone 339 .3 ft 2 71. Denise needs to make a model of a cinder cone volcano for a school play. The cone should be 4 ft tall, the base 4 ft in diameter, and the slant height about 4.48 ft. She is going to build a frame for the cone with wire mesh. How much wire mesh will she need to buy? Surface Area of a Cone where l = slant height = 4.48 ft, r=2ft, h = 4ft SAcone rl r 2 SAcone (2)(4.48) (2) 2 SAcone 8.96 4 SAcone 12.96ft 2 SAcone 40.72 ft 2 85 72. The box for a videotape is 19cm tall, 10.5 cm long, and 2.5 cm wide. It is open on one of the long and narrow sides so that the tape can be put in. How much cardboard is needed to build the box? The net of the box will be made up of 5 faces, two with dimensions of 19 x 10.5, two with dimensions of 10.5 x 2.5 and one with a dimension of 19 x 2.5. The other face with dimensions of 19 x 2.5 will be open for the tape to be put in and will not be part of the surface area figure. 2 x 19 x 10.5 = 399 cm2 Total surface area of the box 2 x 10.5 x2.5 = 52.5 cm2 399 + 52.5 + 47.5 = 499 cm2 19 x2.5 = 47.5 cm2 73. You are building a model of a house. You will make the model as shown below. How much wood dowel will you need to build the frame for the model of the house? If the dimension of the cube is x, the amount needed for the frame will be the dimension of the cube times 12 because there are 12 edges on the cube plus the other 4 edges of the square pyramid on the top of the house. Assuming the triangular faces of the square pyramid are equilateral triangles, each of the 4 edges of the square pyramid are also x. The length of the wood dowel needed to build the frame of the model house will be 16x. 74. A dinnerware factory packs its dinnerware in a pyramid shaped box with a square base of 12 inches by 12 inches and a height of 10 inches. a. How much cardboard is saved using a pyramid design over a square based prism design? b. How much volume is saved? Height of the triangle = slant height Volume of the pyramid a2 b2 c2 1 Vpyramid Bh 10 2 6 2 c 2 3 c 2 136 1 Vpyramid (12 12)(10) c 11.66in 3 Vpyramid 480in 3 Surface area of the pyramid SApyramid Abase 4( Areaface) Volume of the rectangular prism 1 Vprism Bh SApyramid 12 12 4 12 11.66 2 Vprism (12 12)(10) SApyramid 144 279.84 424in 2 Vpyramid 1440in 3 Surface area of the rectangular prism Volume saved with using pyramid SAprism 2(12 12) 4(12 10) The volume of the pyramid is 960 in3 less than the volume of the rectangular prism. SAprism 288 480 768in 2 Surface area saved with using pyramid The surface area of the pyramid is 344 in2 less than the surface area of the prism. 86 Distance and Midpoint Formula 75. Find the distance between (1,4) and (-2, 3). d x2 x1 2 y 2 y1 2 d 2 12 3 42 d 32 12 d 9 1 d 10 76. Find the midpoint between (-2,3) and (4,2). Use a graph to check the result. x1 x 2 y1 y 2 2 , 2 2 4 3 2 , 2 2 2 5 , 2 2 5 1, 2 77. Decide whether the points (3,2), (2,0) and (-1,4) are vertices of a right triangle. To solve this problem, draw the triangle, determine the length of each of the sides, and determine if the lengths of the sides satisfy the Pythagorean Theorem. Let the distance between (3,2) and (2,0) be d1 d1 x 2 x1 2 y 2 y1 2 d1 2 32 0 22 12 22 1 4 5 87 Let the distance between (3,2) and (-1,4) be d2 d2 x 2 x1 2 y 2 y1 2 d2 1 32 4 22 42 22 16 4 20 2 5 Let the distance between (2,0) and (-1,4) be d3 d3 x 2 x1 2 y 2 y1 2 d3 1 22 4 02 32 42 9 16 25 5 Determine if the Pythagorean Theorem holds true for this triangle, proving it to be a right triangle. a2 b2 c2 2 2 5 2 5 52 5 20 25 78. You have a triangle with vertices at A(-3,-2), B(1,2), and C( 2,-2). If you were to draw a new triangle with vertices at each of the midpoints of the sides of the original triangle, how would the perimeters of the triangles compare? The length of the sides of the original triangle are 4 2 , 17 , and 5 respectively. The midpoints of the sides are located at (-1,0), (- ½, -2) and (1½, 0). 17 The lengths of the sides of the smaller triangle are 2 2 , , and 2½. 2 The perimeter of the original triangle, 14.78, is twice that of the perimeter of the new triangle, 7.39. 79. You and your friend go hiking. You hike 3 miles north and 2 miles west. Starting from the same point, your friend hikes 4 miles east and 1 mile south. How far apart are you and your friend? If you want to meet your friend for lunch, where could you meet so that both of you hike the same distance? How far do you have to hike? Assume that the starting point was the origin. The distance between the point (-2,3) and (4,-1) is 2 13 7.2miles You could meet at a point 1 mile east and 1 mile north of the starting point. Both you and your friend would each have to walk 13 3.6miles 88 80. Draw a polygon whose vertices are A(1,1), B(5,9), C(2,8), and D(0,4). Show that the polygon is a trapezoid by showing that only two of the sides are parallel. Use the distance formula to show that the trapezoid is isosceles. Side DC is parallel to side AB because both have a slope of 2. Sides CB and DA are not parallel because DA has a slope of –3 and CB has a slope of 1/3. Sides AD and BC both have a length of 10 , therefore the trapezoid is isosceles. 81. January Young People's LP#1 - New Year‟s Birthdays Amy, 8, bicycle Brent, 12, banjo Jason, 16, digital camera Kristol, 10, telescope Teddie, 14, Nintendo game 82. Jon scores. 83. It needs to be cut. 84. Nothing. Just because Austin got a good haircut does not mean that Jim cut his hair. This is always possible, but nothing can be deduced from the situation. 85. Shaggy is a mammal and a vertebrate. 86. Simply because Adrian's pants are not khaki does not mean that they are not comfortable. Also, even if they are not comfortable, this would not necessarily mean that they are not expensive. There could be comfortable pants that are not khakis, and expensive pants that are not comfortable. Logic Examples 87. 1. If P then Q 2. P 3. Therefore, Q Valid (Modus Ponens) 88. 1. If P then Q 2. Q 3. Therefore, P Invalid This argument form is commonly mistaken as being valid. Notice that even if the premises are true, the conclusion could still be false: Jane could have a dog. 89. 1. If P then Q 2. Not: Q 3. Therefore, Not: P Valid (Modus Tollens) 89 90. 1. If P then Q 2. Not: P 3. Therefore, Not: Q Invalid This is another argument form that is commonly mistaken as being valid. Again, Jane could still have a pet even if she does not have a cat, maybe she has a bird. Her owning a bird is not ruled out by the premises. 91. 1. If P then Q 2. P 3. Therefore, Q Valid (Modus Ponens) Notice that this argument is still valid even though (as far as we know) all the premises (and the conclusion) are, in fact, false. 92. 1. If P then Q 2. Q 3. Therefore, P Invalid This is the same invalid form as argument B. Notice that all the premises and the conclusion are in fact true. Still, the argument is invalid: it is possible for all the premises to be true and the conclusion still be false. You can imagine a world in which the two premises are true, and yet George Bush is not president. Some other Republican could be president. 93. 1. If P then Q 2. Not: Q 3. Therefore, Not: P Valid (Modus Tollens) This is the same argument form as argument C. This seem trickier than argument C since premise (1) in argument G asserts an unlikely relationship between what Joe's favorite color is and whether or not E.T. phones home. What could those two things have to do with one another? They probably have nothing to do with one another. Therefore, premise (1) is probably false. But to check the argument for validity we need to imagine that it is true. So we need to imagine that somehow, for some reason unbeknownst to us, if it is true that E.T. phones home, then it also will be true that Joe's favorite color is blue. 94. 1. Not: P 2. If Q then P 3. Therefore, Not: Q Valid (Modus Tollens) This is the same argument form as argument C and G. The only difference is that the if-then statement is the second premise rather than the first. That's okay, the order of the premises is unimportant for determining validity. Also, don't be fooled by the actual falsity of the premises: IF they were true, the conclusion would have to be true as well. 90 95. 1. P 2. If Q then R 3. Therefore, Not: Q Invalid 96. 1. All x are y 2. All z are y 3. Therefore, x are z Invalid. You can see this by considering an argument of the same logical form that has premises that are easier to imagine being true (because they are true): 1. All humans are primates. 2. All gorillas are primates. 3. Therefore, all humans are gorillas. 97. 1. P 2. Therefore, P Valid Obviously, if "Kate is a lawyer" is true, then it would be impossible for "Kate is a lawyer" to also not be true. But is this because of the logical form of the argument? Well, try uniformly substituting different sentences for 'P' and see what happens. (Remember, whatever you substitute for 'P' must go everywhere there is a 'P'.) However, this argument does beg the question, but that's a different question from the question of validity and invalidity. 98. 1. If P then Q 2. Not: Q 3. Therefore, Not: P Valid (Modus Tollens) Same argument form as C, G, and H. 99. 1. If P then Q 2. Not: P 3. Therefore, Not: Q Invalid Same invalid argument form as in argument D. Even if the premises are true, it is still possible that other life- forms besides human beings have a right to life. It is quite plausible to suppose at the very least that chimpanzees have a right to life. 100. 1. All x are y 2. All y are z 3. Therefore all x are z Valid 101. 1. No x is y 2. All z are y 3. Therefore, no x is z Valid. If it is hard to see why, try drawing a Venn diagram. 91 102. 1. If P then Q 2. P 3. Therefore, Q Valid (Modus Ponens) 103. 1. P or Q 2. Not: Q 3. Therefore, P Valid. 104. 1. P or Q 2. Q 3. Therefore, P Invalid. The premises don't guarantee that Ariel joined the Air force (though he might have.) Note: In logic, the word 'or' is usually understood in its INCLUSIVE sense. You should understand the first premise as saying something to the effect of: "either Ariel joins the air force or Nancy joins the Navy or both". 92 Time Trial #1 - Answer Key 1. D 2. C 3. C 4. B 5. B 6. A 7. B 8. D 9. B 10. Extended response sample Part A. To figure out the amount of paint to purchase to stripe the road, I will start by analyzing the information given in the problem and drawing a diagram. In this situation, we need to paint 7 miles of highway. This distance needs to be converted into yards so that the measurement can be used with the other measurements in the problem that are given in yards. 7 miles times the number of yards in a mile (1760) is 12, 320 yards. From the problem, we also know that on the three lane road there will be two solid stripes that will require 1 gallon for 150 yards and two center stripes that will require 1 gallon for 400 yards as shown in the diagram below. Solid line 150 yd/gal Broken line 400 yd/gal In order to figure out the amount of paint for each stripe, the length of the stripe which are each 12,320 yards will need to be multiplied by the rate given for the number of yards per gallon. For each of the calculations, we will multiply the gallons of paint needed by two since there are two of each stripe. Solid stripe 12,320 yards x 1gal/150 yd x 2 stripes = 61.6 gallons Broken stripe 12,320 yards x 1gal/400 yd x 2 stripes = 164.2 gallons The total amount of paint for the project is 61.6 and 164.2 gallons or 225.8 gallons. 93 Part B To write a formula to figure out the amount of paint needed to restripe any road, the same calculations would need to be done as in Part A, but by defining variables to represent the number of lanes and the number of miles. If we let G be the number of gallons of paint needed, M be the length of the road in miles and let L be the number of lanes, the formula would be as follows: 1760m 1760m g l 1 2 400 150 The first part of the formula allows me to calculate the amount of paint needed for the broken lines. The number of broken lines is always one less that the number of lanes so the multiplier for the number of stripes is L-1. This is multiplied by 1760M, which converts the miles into yards and then divided by 400 or the number of yards that each gallon will paint for a broken line. Similarly, the number of solid lines is always 2, so the multiplier for the solid lines is 2. This is multiplies by 1760M, converting the miles into yards and then dividing by 150 of the number of yards that each gallon will paint for a solid line. Finally, these values are added together to yield the total amount of paint needed to restripe the highway. 94 Solutions for AEPA Time Trial #2 1. C 2. A 3. D 4. B 5. C 6. B 7. A 8. B 9. D 10. D Sample Solution for Extended Response AT = Total area of the yard AG = Grass Area AP = Pea Gravel Area Grass area - Ag Gravel area - Ap 1. The total area of the yard would be found by using the equation A = lw. Since the length of the yard is two times the width, the width can be defined as x and the length as 2x. By substituting these expressions into the area formula for a rectangle, we can find the general formula for the total area of the backyard. At = lw At = 2x x At = 2x2 The area of the grass would be one-fourth of the total area of the yard because the rectangular region of the grass area has one dimension that is ¼ the length but the same width as the total yard. Ag = ¼ (At) Ag = ¼ (2x2) x2 Ag 2 95 The area of the gravel will be the total area minus the grass area. Ap = At – Ag x2 Ap 2x 2 2 4x 2 x 2 Ap 2 2 2 3x Ap 2 2. The ratio of the gravel area to the grass area can be found by dividing the area of the gravel by the area of the grass. 3x 2 Ap 22 Ag x 2 Ap 3x 2 2 3 2 Ag 2 x 1 The ratio of the gravel to grass area is 3 to 1, in other words, the gravel area is 3 times larger than the grass area. 3. If the total area (At) is 5000 sq feet, the rectangular yard would have dimensions of 100 ft x 50 ft. At = 5000 2x2 = 5000 2 x 2 5000 2 2 x 2500 2 x 2 2500 x = 50 ft In the first part of the problem the width of the yard was defined as x and therefore the width of the yard is 50 feet. Since the length of the yard is twice as long as the width (2x), the length is 2 times 50 feet or 100 feet. The area of the grass for this yard would be: x2 Ag where x = 50 ft. 2 50 2 Ag 2 2500 Ag 2 A g 1250 ft 2 Since the area of the grass is 1250 sq feet, that number of square feet of sod would need to be purchased to finish that part of the landscaping. 96

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Clinical Instructor, Journal of Obstetrics, Book Review, School Nurse, Session II, Summer Session, Office Coordinator, Denver, University of Colorado, School of Nursing

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