A security camera is mounted on top of a 15-m-tall pole at the edge of a straight sidewalk

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A security camera is mounted on top of a 15-m-tall pole at the edge of a straight sidewalk. The camera can rotate in a horizontal plane, tracking objects that move along the sidewalk. A courier on an electric scooter travels in a straight line along the sidewalk at a constant speed toward the point on the ground directly beneath the camera. Let x(t)x(t) be the horizontal distance (in meters) between the courier and the pole at time tt seconds, and let θ(t)\theta(t) (in radians) be the angle formed between the camera’s initial forward-facing direction and its current line of sight to the courier. Assume θ=0\theta=0 when the courier is directly in front of the camera and decreases as the courier approaches the pole. (a) Express xx as a function of θ\theta. (b) Derive a formula for dθdt\dfrac{d\theta}{dt} in terms of xx and dxdt\dfrac{dx}{dt}. (c) The courier is 25 m from the pole and moving at 8.0 m/s8.0\text{ m/s} toward it. Compute dθdt\dfrac{d\theta}{dt} (in rad/s) at that instant. (d) Explain whether the magnitude of dθdt\dfrac{d\theta}{dt} will increase, decrease, or remain constant as the courier continues toward the pole, justifying your answer mathematically.

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