Calculus Homework

Question

A rectangular billboard 5 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 theta feet from the billboard. Express theta, the vertical angle subtended by the billboard at her eye, in terms of theta. Then find the distance the cow must stand from the billboard to maximize theta.

Answer 1

Having 2 unknowns with the same variable is extremely confusing so I'm going to rename the distance from the cow to the billboard as "x".

Using angle a:

tan(a) = 5/x so a = tan^-1(5/x)

Using angle b:

tan(b) = 10/x so b = tan^-1(10/x)

Therefore, since θ = b - a, then θ = tan^-1(10/x) - tan^-1(5/x)

To maximize theta, take the derivative and set it equal to zero:

1/(1 + (10/x)²) •(-10/x²) - 1/(1 + (5/x)²) •(-5/x²)

1/(1 + 100/x²)•(-10/x²) - 1/(1 + 25/x²)•(-5/x²)

1/(x²/x² + 100/x²)•(-10/x²) - 1/(x²/x² + 25/x²)•(-5/x²)

1/(x² + 100)/x²)•(-10/x²) - 1/(x² + 25)/x²)•(-5/x²)

x²/(x² + 100)•(-10/x²) - x²/(x² + 25)•(-5/x²)

-10/(x² + 100) - -5/(x² + 25)

-10/(x² + 100) + 5/(x² + 25) = 0

10/(x² + 100) = 5/(x² + 25)

10(x² + 25) = 5(x² + 100)

10x² + 250 = 5x² + 500

5x² = 250

x² = 50

x = √50 ≈ 7.07 feet