optimization question

Question

Two isolated farms are situated 10 km apart on a straight road. The road runs parallel to a highway that is 20 km away. A power company decides to run a wire from the highway to a junction box, and from there, it will run two separate wires of equal length to the two farms. Find the distance from the highway that the junction box should be placed to minimize the length of the wire needed?

Answer 1

Draw a figure!

In order for the distance from the junction box to each farm to be equal, the junction box must lie on the perpendicular bisector of the distance between the farms.

Let x be the distance from the farm road to the junction box.

Then 20-x is the distance from the junction box to the highway.

The distance from the junction box to each farm is sqrt(25+x²)

W, the amount of wire needed is 20-x + 2sqrt(25+x²)

dW/dx = -1 + 2x/sqrt(25+x²) = 0 (at the minimum if there is one).

sqrt(25+x²) = 2x

25 + x² = 4x²

x = (5/3)√3 and the distance from the highway is 20 - x.