Does someone understand this problem? If so, could someone help me please? Thanks!

Question

A rectangular billboard 7 feet in height stands in a field so that its bottom is 8 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x feet from the billboard. Express theta, 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 theta.

Answer 1

The cow is 8-4 = 4 ft below the bottom of the billboard and 4+7 = 11 ft below the top of the billboard.

Thus, if x ft from the sign, the angle to the bottom is cot-1 x/4 and the angle to the top is cot-1 x/11

Thus, θ = cot^-1 x/11 - cot^-1 x/4

dθ/dx = 1/(4(1+(x/4)²)) - 1/(11(1+x/11)²)

Set dθ/dx = 0

1/(4(1+(x/4)²)) = 1/(11(1+x/11)²)

4(1+(x/4)²) = 11(1+x/11)^2

11(4+x)² = 4(11+x)²

176 + 88x + 11x² = 484 + 88x + 4x²

308 = 7x²

x = ±√44

The distance would be |x|, so the solution is √44