Question

A rectangular billboard 7 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.

θ(x) =

Answer 1

To express θ, the vertical angle subtended by the billboard at the cow's eye, in terms of x, we need to use trigonometry. The formula for θ is θ = arctan(8/x). There is no specific distance x0 for the cow to stand from the billboard to maximize θ.

Step-by-step explanation:

To express θ, the vertical angle subtended by the billboard at the cow's eye, in terms of x, we need to use trigonometry. Let's consider a right triangle formed by the cow, the billboard, and the ground. The height of the bottom of the billboard is 12 feet, and the height of the billboard is 7 feet. Since the cow's eye level is 4 feet above the ground, the height of the triangle formed by the cow and the billboard is 12 - 4 = 8 feet.

Using the property that tan θ = opposite/adjacent, we have tan θ = 8/x. Solving for θ, we get θ = arctan(8/x).

To find the distance x0 the cow must stand from the billboard to maximize θ, we need to find the value of x that makes θ maximum. Since the tangent function is maximum when its input is 90°, we can set θ = 90° and solve for x. arctan(8/x0) = 90°. Taking the tangent of both sides, we get 8/x0 = tan 90° = undefined. Therefore, there is no maximum distance x0 for the cow to stand from the billboard to maximize θ.