(Please help!!) The Nearsighted Cow Problem: A Calculus Classic

Question

A rectangular billboard 9 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 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.

Here is the link of the image describes the question: https://d2vlcm61l7u1fs.cloudfront.net/media%2Fe6e%2Fe6ef43ff-5f94-4caf-860d-06b2fcb1df5b%2Fphpx79BUN.png

Questions:

1) θ(x) =

2) θ is maximized when x₀ =

Answer 1

Label the diagram thus:

β is the angle to the bottom of the billboard

γ is the angle to the top of the billboard

θ = γ - β

β = tan (8 / x) → β = arctan (8 / x)

γ = tan (17 / x) → γ = arctan (17 / x)

θ = arctan (17 / x) - arctan (8 / x)

∂θ / dx = -9(x² - 136) / (x² + 64)(x² + 289)

Maximum at x = √136, or 2√34.

Answer 2

I'm figuring from the top of the billboard.

You want the angle between θ and x.

θ=c

x=a

a/sinα=b/sinβ=c

<θx=β (picture)

c^2=a^2+b^2

θ^2=x^2+21^2

θx is max the closer the cow gets to the board.