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

From the description given in the problem, the total amount of wire W in km given as a function of the junction box's distance from the freeway x in km is found to be

W(x) = x + 2 sqrt[(16-x)^2 + 4^2].

Taking the derivative and setting it equal to zero in order to find the minimum yields

dW/dx = 1 - 2(16 - x)/sqrt[(16-x)^2 + 4^2] = 0,

which leads to

2(16 - x) = sqrt[(16 - x)^2 + 4^2].

We can square both sides and solve for x to give

x = 16 +/- sqrt[16/3].

Clearly, since we want the junction box in between the freeway and the road connecting the houses, we only accept the solution

x = 16 - sqrt[16/3].