A power line is to be run to an offshore facility as shown in the figure above. The offshore facility is D = 2 miles at sea and L = 4 miles along the shoreline from the power plant.

Question

A power line is to be run to an offshore facility as shown in the figure above. The offshore facility is D = 2 miles at sea and L = 4 miles along the shoreline from the power plant. It costs $55,000 per mile to lay a power line underground and $95,000 per mile to run the line underwater.

How much of the power line should be run underground, and how much underwater to minimize the overall costs?

Length underground (in miles): ?

Length underwater (in miles): ?

Minimal cost (in dollars): ?

Answer 1

Length underground = 4-x

Length underwater = sqrt(4 + x²)

Cost = 55(4-x)+95 sqrt(4+x²)

Set the derivative equal to zero and solve for x.

The derivative of cost with respect to x is: -55 + 95x/sqrt(4+x²)=0

95x=55sqrt(4+x²)

This results in x=1.42, 4-x=2.58 and z=2.

Please compare your work with mine.

Answer 2

Let's let the shore length be A and the underwater length be B. Then, we can see that the total cost would be 55,000*A + 95,000 * B.

Let's figure out what B is in terms of A. The horizontal distance is going to be 4 - A, and the vertical distance is going to be 2. We can use the Pythagorean theorem to find B in terms of A. From there, substitute that into your cost equation and you'll end up with something in terms of A only. At that point, you should be able to use the standard calculus techniques to find the minimum.