To get the best view of the Statue of Liberty in Figure...

To get the best view of the Statue of Liberty in Figure $4.48,$ you should be at the position where $\theta$ is a maximum. If the statue stands 92 meters high, including the pedestal, which is 46 meters high, how far from the base should you be? [Hint: Find a formula for $\theta$ in terms of your distance from the base. Use this function to maximize $\theta$, noting that $0 \leq \theta \leq \pi / 2 .$ ]

Key Concepts

Angle of Elevation

This concept involves measuring the angle between the horizontal line of sight and the line of sight to an object. In problems like this, the observer's viewing angle is modeled using the arctan function, by relating the height of the object to the horizontal distance from it. Understanding angle of elevation helps in setting up the relationship between the observer, the object’s height, and their separation distance.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the arctan function, are used to determine an angle from a given tangent value. In the context of this problem, they serve as the mathematical tool to translate the ratio of the height of the object to the distance from its base into an angle, which can then be analyzed and optimized.

Differentiation

Differentiation is a key technique in calculus used to find the rate at which a function changes. By differentiating the function that represents the viewing angle with respect to the distance, one can determine where the function reaches a maximum or minimum, which is essential for optimization problems.

Optimization in Calculus

Optimization involves finding the maximum or minimum value of a function subject to certain constraints. In this problem, the goal is to determine the distance from the object that maximizes the viewing angle. This is achieved by setting the derivative of the angle function to zero and solving for the distance, a standard procedure in calculus-based optimization.

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