A monitoring device records the angle (in radians) of a telescopic camera mounted on…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

A monitoring device records the angle θ\theta (in radians) of a telescopic camera mounted on a moving cart. The height of the camera above ground is modeled by h(t)=t2−3t+2h(t)=t^{2}-3t+2 (in meters) at time tt seconds, while the horizontal distance from the origin remains fixed at 10 meters. The angle is related to height by θ(t)=arctan⁡ ⁣(h(t)10)\theta(t)=\arctan\!\bigl(\tfrac{h(t)}{10}\bigr). (a) Calculate dθdt\dfrac{d\theta}{dt} in terms of tt. Simplify the result as a single rational expression. (b) Find d2θdt2\dfrac{d^{2}\theta}{dt^{2}} and evaluate it at t=2t=2. (c) Interpret the sign of d2θdt2\dfrac{d^{2}\theta}{dt^{2}} at t=2t=2 in the context of the camera’s vertical motion. Explain your reasoning verbally without additional calculation.

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