Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to single stake, running from ground level to the top of each post. Where should the stake be placed to use the least wire?

Answer 1

If we place the origin (0,0) at the base of the 12' stake, then the 28' stake has its base at (30,0).

If the length of wire used is

L = sqrt(x^2 + 12^2) + sqrt((x-30)^2 + 28^2)
= sqrt(x^2 + 144) + sqrt(x^2 - 60x + 1684)

dL/dx = x/(sqrt(x^2 + 144) + (x-30)/sqrt(x^2 - 60x + 1684)

dL/dx = 0 when x=9

so, place the stake between the poles, 9' from the short pole, and 21' from the tall pole.