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The Nearsighted Cow Problem: A Calculus Classic. A rectangular billboard 9 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 feet from the billboard. Express , the vertical angle subtended by the billboard at her eye, in terms of . Then find the distance the cow must stand from the billboard to maximize .
The Nearsighted Cow Problem: A Calculus Classic. A rectangular billboard 9 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 feet from the billboard. Express , the vertical angle subtended by the billboard at her eye, in terms of . Then find the distance the cow must stand from the billboard to maximize .
Final answer:The vertical angle subtended by the billboard at the cow's eyes can be calculated using trigonometry, and the distance the cow must stand from the billboard to maximize that angle can be calculated using calculus. The solution is derived by setting the derivative of the angle function to zero and solving for the distance.Explanation:This problem is a classic application of trigonometry and calculus. In the problem,θ, the angle subtended by the billboard at the cow's eye, can be expressed as a sum of two trigonometric angles, namely, arctan(9/x) and arctan(9/(x+9)).Let’s define the height of the billboard as h and the distance between the cow and the billboard as x.Therefore θ = arctan(h/x) - arctan(h/(x+h)). To maximize θ, we can take the derivative of θ with respect to x and set it equal to zero.Using calculus, you can solve for x to find theoptimal distancex the cow needs to stand from the billboard.Learn more about Calculus here:brainly.com/question/35182200#SPJ11...