Precalculus* Name�)9 by cGYCwrJ


									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

     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.

               21     45     74     105    136    166    198    232     259    289     320    351      386
              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

      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
                    70 paces         Drawings not               125'
           P                         to scale                                  50'

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