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\centerline{\bigrm Lori and Jerry's  Trail Optimization Problem}
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\noindent
{\bf Purpose:}
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.

\bigskip

\noindent
{\bf 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 {\bf neatly}.  Be sure to include relevant
graphs
in what you turn in. Your grade will be determined
in part by your presentation.
\bigskip

\noindent
{\bf 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?

\item{1.}Analyze the problem and find a formula for the hiking time in
         terms of the angle of elevation.
\item{2.}Try to transform your answer in Part~1 so you can solve the
         problem by finding the maximum or minimum of a polynomial over
         a range of values for $x$. (You
         may wish to think of the problem as {\bf maximizing} one over the
         time it takes to get to the top.)
\item{3.}Use Mathematica to plot graphs to find the value you are
         seeking in Part~2.
         Try to guess exactly what the correct answer is. (Hint: square
         the decimal approximation you get from the graph.)
\item{4.}Prove your answer in Part~3 is correct and give Jerry
         and Lori the answer to their problem.  (Hint: To prove your
         answer you may wish to translate the graph of the polynomial
         so the point in question moves to the origin.  A
         different approach would be to intersect your graph with a
         horizontal line through the point in question.)
\medskip

\noindent
Note that what you have done in Parts~3 and~4 is to find a turning
point of a particular polynomial. The rest of the project is to
determine turning points for other polynomials.
\medskip

\item{5.}Find the turning points of the function $x^3+4x^2+4x-1$ and
         prove your answer is correct using the same ideas you used in
         Parts~2 and~3.
\item{6.}Find the formula for the turning points of the function
         $x^3+bx^2+cx+d$.
         (Hint: Think about what happened in the two examples above.  If
         you are still not sure what to do, try a few more examples of
         specific polynomials of degree 3 to help you see what to do in
         general.)
\item{7.}{\bf Extra Credit:} Determine an equation that the $x$-coordinates
         of all the turning
         points of a general polynomial
         $a_0+a_1x+a_2x^2+\cdots + a_nx^n$ must satisfy.
\bye
