The Nearsighted Cow Problem: A Calculus Classic.

Question

A rectangular billboard 9 feet in height stands in a field so that its bottom is 10 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands 𝑥 feet from the billboard. Express 𝜃θ, 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.

𝜃(x)=?

 𝜃 is maximized when 𝑥=?

Answer 1

Does this help?

θ = β - α

tan(α) = 6/x so α = tan^-1(6/x)

tan(β) = 15/x so β = tan^-1(15/x)

To maximize, take the first derivative and set it equal to zero and solve. Answers may be either a min or a max so watch it (use first derivative test if necessary).