Two poles are connected by a wire that is also connected to the ground between the poles. The first pole is 3 feet tall and the second pole is 4 feet tall.

Question

There is a distance of 5 feet between the two poles. How far from the first pole should the wire be anchored to the ground to minimize the amount of wire needed? Enter an exact answer or round to the nearest hundredth of a foot.

Answer 1

Let x be the distance along the ground from the 3 foot pole. The wire is the sum of the hypotenuses of the two triangles formed:

L = sqrt(3² + x²) + sqrt(4² + (5-x)²) = sqrt(9+x²) + sqrt(41 - 10x + x²)

Set dL/dx = 0 to find an extremum:

dL/dx = (1/2) * (9+x²)^-1/2(2x) +(1/2)(41-10x + x²)^-1/2(2x-10)

With algebra, you end up with the quadratic 7x² +90x - 225 = 0 and x = 15/7 (the other root is negative). Check against extremes: sqrt(3² + 5²) + 4 and sqrt(4²+5²) + 3 to see if it is a min or max.

Take care.