Purpose: This project will give you an introduction to maximizing and minimizing

Question

Purpose:

This project will give you an introduction to maximizing and minimizing func

This project will give you an introduction to maximizing and minimizing functions. Although you will not learn a general technique, you will use what you know about graphing functions and solving equations to find the minimum of a family of functions. Procedure: You are to follow the steps below. You do not need to do the steps in exactly the order indicated as long as you cover everything you are asked to do. After completing the outline below you should write your results and calculations neatly. Be sure to include relevant graphs in what you turn in. Your grade will be determined in part by your presentation. Project Outline: Jerry and Lori have been asked to design a trail which goes form a camp site which is at an elevation of 8000 feet up a very steep mountain to an elevation of 10,000 feet. They plan to make the trail spiral up the mountain, which happens to be shaped approximately conical. They are therefore able to give the trail a constant slope all the way to the top. They also decide that they wish to make the trail in such a way that the time it takes for most hikers to get to the top is minimized. After consulting some hiking experts they found that the speed a hiker hikes is proportional to the square of the cosine of the angle of elevation of the path. What angle of elevation should they use to minimize the time it takes to hike the trail? Analyze the problem and find a formula for the hiking time in terms of the angle of elevation. Try to transform your answer in part 1 so you can solve the problem by finding the maximum ar minimum of a polynomial over a range of x values (you may wish to think of the problem as maximizing one over the time it takes to get to the top.)

Explanation / Answer

The distance they travel is 2000/sin (theta). This is because sin theta = vertical distance/ total distance

Rate they hike is k cos^2 (theta).

Distance = rate times time

so time is distance divided by rate

time = 2000/[sin(theta) k cos^2(theta)]

This is the answer to 1

In 2 we see that 2000 isn't changing and k isn't changing, so that to minimize t, we maximize the denominator

sin(theta) cos^2(theta) is what we want to maximize. Are we all thinking trig identities??

sin(theta) (1 - sin^2(theta)) is maximum when it isn't increaseing any more or decreasing anymore and it is positive. When you are at the highest; you aren't going any higher and you aren't on your way don yet.

So, I graphed this and find 0.6155 radians

this is 35.52 deg

So then I did it in Microsoft Excel, the last column shows the time, and we see that time is the smallest, when x is 0.62 radians or 35.52 degrees

radians, sin, cos, cos^2, sin*cos^2, degrees, 2000/sin, cos^2, time

0.6 0.564642 0.681179 0.384623 34.37747 3542.064 0.681179 5199.903 0.61 0.572867 0.671823 0.384865 34.95043 3491.209 0.671823 5196.621 0.62 0.581035 0.662398 0.384877 35.52338 3442.132 0.662398 5196.471 0.63 0.589145 0.652908 0.384658 36.09634 3394.751 0.652908 5199.429 0.64 0.597195 0.643358 0.38421 36.6693 3348.987 0.643358 5205.483

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