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Precalculus* Name Spring 2009 Per Date Final Version DUE: Tuesday May 12, 2009 TRIG PROJECT READ and FOLLOW these directions carefully. Point(s) will be deducted any time directions are not followed carefully. 1. Retype each problem on a word processor before working it. Then show all work on the problem. The work may be shown by hand, following the guidelines below. You may re-word problems to create your own story, but keep the mathematical content the same as in the problem statement. Note: This will be posted on Mr. K's home page as the problems are given to you so you can download, copy, and paste if you want to do so. 2. Put only no more than one problem on a page with problems on only one side of a page. Be sure to leave a wide enough left margin for the type of folder you are using. 3. Include a title page and a binder or report cover similar to what you would use for a paper in another subject. Arrange problems in the order given when handing in your project unless dictated otherwise by your creative efforts. You may use a folder with fasteners in it. However, a plain manila file folder or pocket folder is not acceptable. Include your title, name, period, and date. 4. Draw a diagram for each problem. This includes a REDRAW if the conditions change. Attempt to make diagrams as nearly to scale as possible. This may make problems easier to solve and check the reasonableness of your answers. Be sure to label all quantities represented in the diagram whether they are knowns or unknowns. (or lose points each time) Certain diagrams may be downloaded, buy they may not be to scale. 5. Show all steps in your solution in a neat and organized manner. Proceed from step to step VERTICALLY with a brief written explanation to the right of each step (not necessarily each calculation). NOTE: THIS PART HAS BEEN NEW SINCE 2007!!! I need to be able to follow all of your work easily without having to hunt for things. This includes how you got your angles, even if just “doing the 180-thing” from Geometry. Do not leave anything to my imagination - I will have none! 6. Use the following guidelines for rounding answers unless stated differently in a problem. You may round intermediate answers to four (4) decimal places or keep all decimals. In final answers round lengths to the nearest hundredth unless stated otherwise. If the angle(s) in a problem are given in degrees, minutes, and seconds (DMS), give your answer(s) for other angle(s) in DMS also. Otherwise, round to the nearest hundredth of a degree. NOTE: If the answer to one part of a problem must be used to answer a subsequent part, you should use the UNROUNDED intermediate answer or a 4-place decimal intermediate answer. It is best to use the STO button on the calculator to keep all the decimals. Points will be deducted EACH time these guidelines are not followed. All of your answers should match mine. 7. Be sure answers are indicated or marked clearly. This includes important intermediate answers. I must be able to find everything! Trig Project 2009 Pg. 2 8. Label all answers with appropriate units such as degrees, DMS, feet, miles, knots, etc. as dictated by the problem if they are given. If you need to give a heading or bearing, you may use either notation which we have used, as long as your numbers are correct. 9. Points will be included in the grade for following directions, rounding/accuracy, neatness, thoroughness, creativity/uniqueness, and appropriateness of methods used. 10. You must include a brief list of the names of people you worked with or received HELP from on EACH PROBLEM SEPARATELY, not just at the end of the project. If you did not receive any help on a problem, tell me that also. There is to be no copying of problems, but you may HELP one another. Any deviation from this will be severely penalized. Include the names of any teachers or any other non-students who helped. THE PROBLEMS 1. Gotta Make a Lake! A plot of land has been donated to create a manmade lake. If the only dimensions of the plot are as shown in the diagram below, answer the following questions: : a) Find the approximate area of the plot of of land, to the nearest square foot. b) Assume that 5 % of the area above will not be actual lake, but some of it will be land, due to the angles of the pentagon given. If the average depth of the lake is to be 5 ft., find the volume of the lake to the nearest ft3. c) How many gallons of water will the lake hold? Use the conversion factor that one cubic foot is equal to 7.48 gallons. Give your answer to the nearest gallon. d) Periodically, the lake must be treated with an anti-algae chemical to prevent “pond scum” from forming on the top. If 8 fluid ounces of the chemical will treat one cubic yard of water, how much of the chemical will be needed for each treatment. Give your answer to the nearest gallon. (assume you cannot store any of the chemical for the next treatment, so don’t buy ahead) e) If the chemical in part d) above may be purchased for $2.19 per gallon, $9.99 for 5 gallons, or $89.99 for a 50-gallon drum, what is the most economical way to purchase the chemical, AND how much will it cost?. (nearest cent) Note: A cleaner drawing is included at the end of the project. Trig Project 2009 Page 3 2. Locating Lost Treasure! While scuba diving off Wreck Hill in Bermuda, a group of five entrepreneurs discovered a treasure map in a watertight cask on a pirate schooner in 1747. The map directed them to an area of Bermuda now known as The Flatts. The directions on the map read as follows: 1) From the tallest palm tree, sight the highest hill. Drop your eyes vertically until you sight the base of the hill. 2) Turn 40º clockwise from that line and walk 70 paces to the big red rock. 3) From the red rock walk 50 paces back to the sight line between the palm tree and the hill. Dig there. The five entrepreneurs believed that they found the red rock and the highest hill, but the tallest palm tree had long since fallen and disintegrated. It occurred to them that the treasure must be located on a circle with radius 50 “paces” centered at the red rock. They decided a “pace” must be about a yard. Answer each of the following. a) If the five entrepreneurs dig a trench along the entire circumference, how long will it be? b) Determine a plan to locate the position of the lost palm tree, and write out an explanation of your procedure for the entrepreneurs. Include a sketch. c) One possible solution to find the treasure follows: From the location of the palm tree (found in your procedure above), turn 40º counterclockwise from the red rock toward the hill, then go until you hit the circle traced about the red rock. Verify this solution (show sketch and why it would work). d) Unfortunately, they found no treasure at the first place they dug. Find the actual location of the treasure. ALSO tell how far the treasure is from the original palm tree’s location? Be sure this is shown in a new sketch or the one above, and be sure to justify it. e) Find the area of the triangle determined by the first spot they dug, the palm tree location and the red rock, and of the triangle determined by the second spot they dug, the palm tree location and the red rock. Note: A cleaner drawing is included at the end of the project. Trig Project 2009 Page 4 3. Gone Fishin’! Bud the Fisherman is going fishing on a very large lake in Minnesota. He takes off from shore toward his favorite fishing spot which is 30 nautical miles away at a heading of 322º24' But alas, an unusual wind is blowing at a heading of N 74º50' W causing the boat to be pushed off course. If the boat’s speed over still water would be 12 knots and the wind is pushing it off course at a rate of 3.5 knots, find each of the following: a) If Bud does not compensate for the wind, find his actual speed (Speed Over Ground or SOG) and actual heading (Course Made Good or CMG) b) After traveling for the usual 30 nautical miles, his first mate, Ruth, realizes that something is awry, and tells Bud to stop. They drop anchor and come to the realization that they are off course. Find their distance and bearing to the favorite fishing spot from this location. c) Unfortunately, Bud and Ruth cannot get the boat started. After sending a distress call to their sons, Greg, Larry and Jerry, “The Boys” take of from the favorite fishing spot to go rescue their parents in their boat. If they take the wind into account, find the course they must take in order to arrive at Bud’s boat. d) At a speed of 4 knots, how long will it take them to arrive? e) Fortunately, Bud had bait with him in the boat. (one always carries extra bait when fishing in the Land of 10,000 Lakes). While waiting for “The Boys”, Bud and Ruth each catch several bullheads (a type of catfish), some walleye, and one lonely perch, so they will have enough to clean and eat. Questions: 1) What is the biggest fish you have ever caught? 2) Did you bait your own hook? 3) Did you take your fish OFF the hook yourself? See next page for #4 Trig Project 2009 Page 5 4. Weather or Not! The table below has average monthly temperature data for Cleveland Hopkins International Airport. To make your work easier, I have selected a day of the month for which the daily average is about the same as the monthly average. The day number represents the day of the year (out of 365) with the last value going into the next year to complete your eventual graph. Use this information to do or answer the questions following the table. Day 21 45 74 105 136 166 198 232 259 289 320 351 386 (d) Temp 25.7 28.4 37.5 47.6 58.5 67.5 79.9 70.2 63.3 52.2 41.8 31.1 25.5 (T) a) BY HAND, find the sine equation that could approximate the data in the table. Show how you got all the pertinent information for your equation. I’ll assume nothing here! Use days, d, as your independent variable, and the temperature, T, as your dependent variable, for an equation such as T(d) = . . . . Write your equation in the form we used in Chapter 8 as follows: y = A sin(B(x – h)) + k. Show how you got each of the values you used to make the equation. b) Use the graphing calculator to create a scatter plot of the data and a sine regression equation like we did in class. You should already know how to do this – or you can download the instructions from my home page. You’ll need RADIAN mode. Be sure to use an appropriate window. INCLUDE the following grapher screens in your project, obtained from the Graph Link or Virtual TI (in no particular order): 1) the scatter plot AND regression equation on the same graph, 2) the window dimensions you use, and 3) The actual regression equation you find in the Y= menu, including your last name typed in as a function (after you do the graph). Note: you’ll have to scroll way down or up 1 to find SinReg in the STAT - CALC menu. c) Answer the following based on the grapher screens in part c above. If you use the grapher for part 1), show (an) appropriate grapher screen(s). 1) Based on the regression equation, what would be the expected average temperature on May 8, the date of the release of the new Star Trek movie? Show how you got the number of day you used. 2) On which days of the year is the approximate average temperature 32º? You may want to know this for gardening purposes. Give your answer(s) as a date(s), not just the number of the day of the year. NOTE: You should solve this one BY HAND, showing each step, until the very last step! You may round the values in your regression equation to 4 decimal places rather than use all of the digits the calculator gave you. (Over for more on #4) Trig Project 2009 Page 6 #4 Continued: Grapher Screens required for #4: Part B: - Scatter Plot and graph of Regression Equation on same graph - Window dimensions used - Y= screen with Regression Equation WITH YOUR LAST NAME on it Part C: - Graph with necessary adjustments, including the point(s) of intersection (unless you do all parts by hand) This is for part 1 only. Part 2 is all by hand until the last step! NOTE: You can save time by using the following syntax on the calculator to put the regression equation into the Y= menu automatically: SinReg L1, L2, Y1 With L1, L2, matching your lists, and Y1 matching whatever your equation will be named in the Y= menu. To get L1 and L2 press 2nd and the numbers 1 and 2. For Y1, press VARS, then YVARS, then FUNCTION, then Y1 and press ENTER. After thinking for several seconds, the numbers for the regression equation will appear on the screen AND it is automatically placed into the Y= menu as Y1. Remember to get the SinReg command, press STAT, CALC, and scroll to SinReg. 5. How Close?! In Calculus, there are things called “limits” which are very important in developing ideas in Calculus and other math courses. We will learn about some of them this year. Answer the following questions about limits to get you thinking about them. a) If you start out 16 feet from a wall, and in successive “jumps,” end up half as far from the wall, theoretically, when do you reach the wall? Explain your answer. Answer the following using your calculator in radians. You do NOT need to show any screens from the grapher. b) Use the table (optional) on your calculator to tell what happens to y = x as x gets very close to zero. Use x-values of 0.1, 0.01, 0.001. This is the limit as x approaches zero. We use the following notation – include this and your answer in your project. Show your decimal values also. More details on using the table (if you choose to) are given below. lim ( x ) ? x 0 Trig Project 2009 Page 7 Problem #5 (continued) c) Do the same as above for y = sin(x) as x gets very small. Use x-values of 0.01, 0.001, 0.0001 (all in RADIANS) Write a limit statement like above for your answer, and include decimals. sin x d) According to your two answers above, what would be lim ? Give this in its original x 0 x form. Can you evaluate this? Give you answer as a limit statement like you did above. Now go to part e) below. e) Use the TABLE on your calculator to find the limit above in part d). Go to 2ND WINDOW to get to the TBLSET feature. Then choose a small starting x value (TblStart), and a smaller ΔTbl, and use Auto for the last two settings. Go to the table (2ND GRAPH) and see what happens near and at x = 0. You should also look at the graph. Does it look like there is a point on the graph? Is there? Why or why not? What is happening to the y-value as you get CLOSE to x = 0, but don’t actually get there – or what does y get CLOSE to? Be sure to do or answer each of the following, but you do not need to show any graphing calculator screens: - On the graph, does it look like there is a point there? - On the table, what happens? Tell what you used in your TABLE settings! - As x gets close to zero, what does y get close to? - Write a limit expression for this like the ones above. 1 cos x f) Do the same as you did above for the following limit: lim answering all the same x 0 x questions that were asked in parts b), c), d), and e). 6. Of all the trigonometry we have done since Chapter 7, what was your favorite thing to do or favorite type of problem and why? Give an example of this type of problem. You do not have to provide a solution. 50 paces 100' 70' ? 100º R 50º ? 50' 70 paces Drawings not 125' necessarily P to scale 50'