how to minimize the total amount of wire needed

Question

Another friend of the Lee family, the Bloch family, has a cottage that is situated 8km down the road from them (it’s a straight road). The road runs parallel to the main highway that is 16km away. A power company needs to run a wire from the highway to a junction box (situated somewhere between the highway and cottage road). From there, it will run two separate wires of equal length to both the Lee and Bloch family cottages. Determine the distance from the highway that the junction box should be placed to minimize the total amount of wire needed.

Answer 1

You need to make a figure which will look like an inverted Y.

The leg of the Y = 16-x and the distance from the crotch of the Y to the country road is x.

The distance from the crotch to each house is sqrt(16+x²)

The length of wire w=16-x+2sqrt(16+x²)

dW/dx= -1+[2x/sqrt16+x²)]=0

x=(4/3)sqrt(3)

You can find 16-x which is the distance from the main highway.