Calc Question math

Question

A rectangular billboard 8 feet in height stands in a field so that its bottom is 12 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x feet from the billboard. Express delta, the vertical angle subtended by the billboard at her eye, in terms of x. Then find the distance x0 the cow must stand from the billboard to maximize delta.

delta x = delta is maximized when x0=

Answer 1

δ(x) = tan^-1(16/x) - tan^-1(8/x)

δ'(x) = 1/((16/x)²+1)(-16/x²) - 1/((8/x)²+1)(-8/x²) = (1024-8x²)/(x⁴+320x²+16384)

Set δ'(x) to zero and solve for x

8x²=1024

x = √128 = ±11.313 feet (use the positive value)

x=11.313 feet