powerline.html

Power Line through the Park

For this project, you will need to know how to use the derivative to determine local extreme values of a function. Since this involves finding values of x where the derivative is zero, you will also need to solve equations. Sometimes it is convenient to solve equations algebraically with the "solve" command. At other times it is convenient to solve equations numerically with the "fsolve" command.

Finding Extrema of Functions:

You know how to find maximum and minimum values of a function on a closed interval [a,b]---Find all points of the graph at which the derivative either does not exist or is zero (critical points), then compare the height of the function at each of the critical points with its height at and at . The largest of these values is the maximum value of the function on the interval and the smallest is the minimum. You can use a graph plotted by Maple to assist you in determining which of these points is the maximum point and which is the minimum point. Here is an example:

To find the maximum and minimum values of the function f ( x ) = on the closed interval [0,2]:

> restart;f:=x->x^3-2*x+1;

Find the derivative of f:

> D(f)(x);

Find critical points of f:

> fsolve(D(f)(x)=0,x);

You see that only one of the critical points lies in the interval [0,3]. So one of f(0), f(.8164965809), and f(2) is the maximum value and one is the minimum value of f on this interval. Look at the graph of f(x):

> plot(f(x),x=0..2);

You see from the graph that the maximum point lies at x=2 and the minimum point is at x= . So the minimum value of the function is f( ) and the maximum is f(2):

> {f(.8164965809),f(2)};

So the maximum value of f on [0,2] is f(2)=5 and the minimum value is f(.8164965809)= -.088662108.

>

Project: Power line

Your Problem:

A rectangular city park is one mile by two miles and is located in the center of Oakdale. Oakdale Power and Light must run a power line from the northwest corner A to the southeast corner B of the park. You, as city engineer, must determine the least expensive installation of the power line. Here is a diagram of the park:

Though overhead power lines are less expensive than underground lines, city ordinances prohibit stretching overhead line through the park, so that although the shortest distance runs directly from point A to point B you may not assume that this gives the cheapest line. You plan to run overhead line a distance x along the south side of the park from point B to point C, then use underground line directly from point C to point A at the southeast corner of the park. The low bid for purchase and installation of underground line is 100 thousand dollars per mile. Suggestion: Work in units of thousands of dollars or hundreds of thousands of dollars in order to keep your numbers small.

Case 1: Suppose the low bid for purchase and installation of the overhead line is 90 thousand dollars per mile, not much less per mile than the cost of the underground line, which is 100 thousand dollars per mile. Determine the value of x for the cheapest installation. Determine the total cost for this value of x. How does this compare with the cost of running the cable overhead around the edge of the park?
Case 2: Suppose the low bid for purchase and installation of the overhead line is 75 thousand dollars per mile, considerably less than the cost of the underground line, which is 100 thousand dollars per mile. Determine the value of x for the cheapest installation. Determine the total cost for this value of x. How does this compare with the cost of running the cable overhead around the edge of the park?
Case 3: Suppose the low bid for purchase and installation of the overhead line is 50 thousand dollars per mile, much less than the cost of the underground line, which is 100 thousand dollars per mile. Determine the value of x for the cheapest installation. Determine the total cost for this value of x. How does this compare with the cost of running the cable overhead around the edge of the park?
Suppose the cost per mile of the overhead line is k thousand dollars. For each of the values k = 10,20,30,...,90,100, determine the minimum cost of laying the power lines and describe the manner in which the power lines should be run in order to achieve the minimum cost.

Extra Credit

You are preparing a report for the mayor of Oakdale in advance of opening the bid for the less expensive overhead line.Let the cost per mile of the overhead line k thousand dollars where k is between 0 and 100. The cost per mile of the expensive underground remains at 100 thousand dollars per mile. Determine the values of k for which it is cheapest to wire overhead around the edge of the park. Determine the values of k for which it is cheapest to lay cable underground directly from A to B. For the remaining values of k determine the value of x for which the total cost is minimal. Write your conclusion so concisely that the mayor can determine the lowest bid for the overhead line and immediately announce the design of the pwer line without having to do any further calculations.

The Most Common Maple Commands

Academic Honesty Statement:

Place the following statement (by copying and pasting) at the end of your report and sign it in ink. Your instructor will not grade your report unless this signed statement appears at the end of your report.

I understand that I may work with others if I give them credit in this statement. I also understand that I am required to write my report--that to copy all or part of someone else's report or to allow someone else to copy all or part of my report constitutes plagiarism, which is a serious violation of academic honesty.

I worked with (replace this parenthetical remark with first and last names of those with whom you worked) on this project. I wrote my own report. I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.

Signed: