A rectangular billboard 6 feet in height stands in a field so that its bottom is 9 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x feet from the billboard.

Question

Express θ, 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 θ.

Answer 1

From the cow's eye, the top of the billboard is 15' high, and to the bottom 9'.

So the angle from the cow's eye, 4' high, Θ = arctan(11/x) - arctan(5/x)

To maximize this angle, taking the derivative and setting to 0, yields

11/(x² + 121) - 5/(x² + 25) = 0

Solving yields x = √(55) = 7.4 feet away from the billboard.