You are working for a power plant that is located on a highway that runs east to west.

Question

The town of Cortez is located 38 miles east of the power plant, and 5 miles north of the freeway. The town of Joplin is located 32 miles east of the power plant, and 9 miles south of the freeway.

To connect power to the two towns, the power plant wants to build one junction station along the freeway then run power out to the two cities from the junction station. The picture below shows the general layout (not to scale), where x is the length of power line run along the freeway (the length from the power plant to the junction at point P). Your boss has asked you to determine where the junction station should be built.

1)

Find a formula for the length of power line needed to run from the junction at point P out to Cortez as a function of x.

2)

Find a formula for the length of power line needed to run from the junction at point P out to Joplin as a function of x.

3)

Find a formula for total length of cable needed for running all three length of cable

4)

Use your calculator to find the value of x that minimizes the total length of cable needed.

Answer 1

I'm guessing the picture ;looks like this:

Then, using the Pythagorean Theorem, d_C(x) = √(38 - x)² + 5²) or d_C(x) = √(38 - x)² + 25) and d_J(x) = √(32 - x)² + 9²) or d_C(x) = √(32 - x)² + 81)

The total distance (d_T) is found by d_T(x) = x + √(38 - x)² + 25) + √(32 - x)² + 81)

When you graph this function you get a graph with a minimum value at x = 30.475 miles