FInd the Nearsighted Cow problem

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 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

angle A formed by a right triangle with base length x and height = 12-4 = 8

has tangent = 8/x, so A=arctan(8/x)

angle B formed by another right triangle with base x and height 20-4 = 16

has tangent 16/x, with B= arctan(16/x)

angle for the near sighted cow's vision of the board is

B-A=arctan(16/x)-arctan(8/x)

take the derivative of B-A, with respect to x, set=0,solve for x = distance from the cow to the board that maximizes B-A

(B-A)' = 8/(64+x^2) - 16/(256+x^2)=0

1/(64+x^2)= 2/(256+x^2)

2(64+x^2)= (256+x^2)

128+2x^2 =256+x^2

x^2 = 128

x=8sqr2

= about 11.314 feet

B-A - arctan(sqr2)-arctan(1/sqr2)

B-A= about 19.471 degrees= angle maximizing cow's billboard view

this all assumes x is measured from the cow horizontally to a point vertically below the bottom of the billboard. if not,then add another wrinkle to the problem

if the cow is too nearsighted then the answer is No solution, Null set, DNE, { }, Does Not Exist