Finding Minimum Point

Question

Two poles 20 and 28 feet high respectively are 30 feet away. It must be fastened with cables fixed at a single point, from the ground to the ends of the points. Where are the cables to be fixed so that the amount of cable (x + y) to be used is minimum?

Answer 1

Let d is distance to the pole 20 feet high. Then length l of cable l(d) = (400 + d²)^1/2 + (784 + (30 - d)²)^1/2

l(d) = (400 + d²)^1/2 + (1684 - 60d + d²)^1/2.

l'(d) = d/(400 + d²)^1/2 + (d - 30)/(1684 - 60d + d²)^1/2 = 0;

d(1684 - 60d + d²)^1/2= (30 - d)(400 + d²)^1/2;

d²(1684 - 60d + d²) =(30 - d)²(400 + d²); d⁴ + 1684d² - 60d³ = 360000 + 900d² - 24000d - 60d³ + 400d² + d⁴ = 0; 384d² + 24000d - 360000 = 0; 32d² + 2000d - 30000 = 0; 2d² + 125d - 1875 = 0; d = (- 125 ±√30625)/4;

d = (- 125 ±175)/4; d = 12.5 feet