Precalculus* Name®)9¯ ®):¯



Honors Trigonometry/Pre-Calculus Project (2013)

Due Date: Friday May 17, 2013

Name: _________________________________ Period: _______ Date: _____________

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: The project will be posed along with hints and tips at crunchymath. so you can download, copy, and paste if you want to do so.

2. This project may be completed independently, with a partner, or with at most a group of three.

3. Start each new problem on a new page, using only one side of the page. Multiple parts of a problem may be on the same page. Be sure to leave a wide enough left margin for the type of folder you are using.

4. 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. Note: If you are working with a partner or with a group of three, the title page should have all names on it. You need only to turn in one project for the group.

5. 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 representing the diagram whether they are knowns or unknowns (or lose points each time). Certain diagrams may be downloaded, but they may not be to scale.

6. 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). 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!

7. 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.

Trig Project 2013 Pg. 2

8. Be sure answers are indicated or marked clearly. This includes important intermediate answers. I must be able to find everything!

9. 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.

10. Points will be included in the grade for following directions, rounding/accuracy, neatness, thoroughness, creativity/uniqueness, and appropriateness of methods used.

11. 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 beginning or 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.

Trig Project 2013 Page 3

THE PROBLEMS

1. Holy Moley! . . . . A certain teacher, who shall remain nameless, has a plot of land that has been overrun with moles. In order to drive them moles away without killing them, he will spread organic “Mole-Not” on his property to kill the grubs on which the moles feed. But he needs your mathematical expertise to help! The plot of land with dimensions (in feet) shown in the diagram below. Answer each of the following questions.

a) Find the approximate area of the plot of

of land, to the nearest square foot. ALSO

how many acres is this (nearest hundredth)?

b) Assume that 8.5 % of the area above is covered

by a trees, and landscaping, etc., and will

not need to be treated. How many square feet

does this leave to be treated?

c) If “Mole-Not” costs $27.99 for a bag that covers 15,000 sq. ft, and 9.99 for a bag that covers 5,000 sq. ft., what is the most economical way to buy Mole-Not? Note that you do not want too much left over since it is dangerous to store. Give the number of each size of bag and the cost.

d) The certain teacher wants to divide the land into two sections with the areas being as close to one another as possible, by constructing a fence along one of the diagonals of the land. Between which two vertices should the teacher construct the diagonal? JUSTIFY your selection of which diagonal to use! This may require finding extra areas at some point.

e) If the certain teacher wants to put fence around the perimeter which costs $2.25 per linear foot, and the fence along the diagonal costs $3.50 per linear foot, how much will the total cost of the fence be?

Trig Project 2013 Pg. 4

2. Off on a Tangent? Three circles of radii 12 inches, 22 inches, and 31 inches are externally tangent to each other. Find each of the following (in whatever order is necessary).

a) Find the measure of each angle of the triangle.

b) Find the area of each circle.

c) Find the area of the triangle.

d) Find the area of the region between the circles

that is inside the triangle.

e) What percentage of is the area you found in part (d) of

the triangle, and of each of the 3 circles? Give each of

your four answers to the nearest hundredth of a percent.

f) Find the total of the arc lengths of the three portions of the circles that are inside the triangle.

Trig Project 2013 Page 5

3. Where's He Headin'? Two airplanes take off at the same time from different runways at the same airport. The larger plane’s speed (without any wind) is 420 mph, and the smaller plane’s speed (without any wind) is 270 mph. The wind affecting both planes is at a speed of 36 mph at a heading of 24º. Answer each of the questions below.

a) If the larger plane is flying at a heading of 324º (without the wind), find BOTH the plane’s actual heading (also known as its “true course” or “course made good”), and it’s actual speed of travel (also known as its “speed over ground”). Remember that these are different than the plane’s intended course and speed because of the wind. NOTE: Do calculations for parts a) and b) for one hour.

b) If the smaller plane is flying at a heading of 78º, find its course made good and its speed over ground.

c) If each plane has been flying for 1.5 hours, find the distance and bearing from the smallr plane to the larger plane at this time.

d) If there is another airport which is 840 miles from the planes’ point of origin at a heading of 28.5º, find the distance and heading from ONE of the planes to the second airport. You do not need to find this for both planes.

Trig Project 2013 Pg. 6

4. [pic]More Tangents??? In this problem you will develop a formula for the slope of the tangent line to the sine curve at an arbitrary point (x, y). In other words, you will derive a formula for the derivative of the function[pic]. The first approach will be graphical, the second will be numerical, and the third will be algebraic.

a) Graphical Approach Use a graphing utility to graph the function [pic] as shown below. Make sure you are in radian mode. Plot your estimates of the slope of this curve at various x-values. For instance, some values of the slope to the right of the sine curve have been plotted (see below). The slope is approximately 1 at x = 0, 0.8 at x = .5, and 0 at x = 1.5. After you have plotted at least 20 points, connect them with a continuous curve. Do you recognize the curve?

[pic] [pic]

b) Numerical Approach The slope of the tangent line is given by the formula

[pic]

The difference quotient [pic]is a good approximation of the slope when h is small. If you choose h = 0.01, you have the following approximation for the slope of the sine curve at x.

[pic]

Use the table feature of a graphing utility to complete the table below. Plot these points and compare your results with part (a).

Trig Project 2013 Pg. 7

[pic]

c) Algebraic Approach To calculate the slope of the tangent line to [pic] algebraically, you need two trigonometric limits. (See part (d) and (e) below)

Limit 1: [pic] Limit 2: [pic]

Use these limits to find a formula for the slope of the sine curve.

d) Use a graphing utility to estimate Limit 1 in part (c) above graphically and numerically. (Sketch the graph and show the tables of values)

[pic]

e) Use a graphing utility to estimate Limit 2 in part (c) above graphically and numerically.

(Sketch the graph and show the tables of values)

[pic]

f) Follow the steps used in parts (a), (b), and (c) above to develop a formula for the derivative of the function [pic]. Is it true that the sine and cosine functions are each other’s derivatives? If not, which is the derivative of the other?

g) Amusement Park Ride An amusement park ride is constructed such that its height h in feet above ground in terms of the horizontal distance x in feet from the starting point can be modeled by

[pic]

The formula for the derivative of [pic] is [pic] Use this to find the derivative of [pic]. Find [pic]

The End (whew!)

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150º

120º

140º

125º

80º

240

195

158

134

U

N

M

O

L

E

(not to scale – but close!)

B

A

C

(not to scale)

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