The Nearsighted Cow Problem: A Calculus Classic.
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 I feet from the billboard. Express (x), the vertical angle subtended by the billboard at her eye, in terms of 2. Then find the distance the cow must stand from the billboard to maximize 0(x).
θ(x) =
(x) is maximized when
The Correct Answer and Explanation is :
To find the vertical angle (\theta(x)) subtended by the billboard at the cow’s eyes, we define two key angles:
- Upper angle: The angle between the cow’s line of sight to the top of the billboard and the horizontal.
- Lower angle: The angle between the cow’s line of sight to the bottom of the billboard and the horizontal.
Step 1: Define the angles
Using trigonometry, the tangent function helps us determine these angles.
- The top of the billboard is at a height of ( 10 + 9 = 19 ) feet.
- The bottom is at ( 10 ) feet.
- The cow’s eye level is at ( 4 ) feet.
- The distance of the cow from the billboard is ( x ).
The angles are:
[
\theta_1 = \tan^{-1} \left( \frac{19 – 4}{x} \right) = \tan^{-1} \left( \frac{15}{x} \right)
]
[
\theta_2 = \tan^{-1} \left( \frac{10 – 4}{x} \right) = \tan^{-1} \left( \frac{6}{x} \right)
]
Thus, the vertical angle subtended by the billboard at the cow’s eye is:
[
\theta(x) = \tan^{-1} \left( \frac{15}{x} \right) – \tan^{-1} \left( \frac{6}{x} \right)
]
Step 2: Maximizing (\theta(x))
To find the distance ( x ) that maximizes (\theta(x)), differentiate:
[
\frac{d}{dx} \left[ \tan^{-1} \left( \frac{15}{x} \right) – \tan^{-1} \left( \frac{6}{x} \right) \right]
]
Setting the derivative to zero and solving numerically, the optimal distance is:
[
x = 10.6 \text{ feet} \ (approximately).
]
Thus, the cow must stand about 10.6 feet from the billboard to maximize her view.
Now, I will generate an image illustrating this problem.
Here is the generated diagram illustrating the Nearsighted Cow Problem. Let me know if you need any modifications or further explanations!
