(Highschool) AP Calculus, Optimization

Question

Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation.

Centerville is located at (11,0) in the xy-plane, Springfield is at (0,9), and Shelbyville is at (0,-9). The cable runs from Centerville to some point (x,0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer. To solve this problem we need to minimize the following function of x:

f(x) = ______

We find that f(x) has a critical number at x = ___

To verify that f(x) has a minimum at this critical number we compute the second derivative f′′(x) and find that its value at the critical number is ___, a positive number.

Thus the minimum length of cable needed is ___.

Answer 1

Here is a video I made to illustrate what is happening in this problem (notice how the amount of cable is dependent on where the point (x,0) is):

https://drive.google.com/file/d/1Y0j770fXqr_Fg6DHbJCUp4YdjJVxb7xf/view?usp=sharing

The critical piece here is the formula for the total distance (named D in the video). It is found by summing the distances of each town from the point (x,0) using the distance formula. Definitely do this on your own and verify my algebra. Then you just take the derivative and set to zero, solve for x, verify that it is in fact a minimum with second derivative.

Visualization can be a big help in these types of problems. Always try to draw a diagram.