A drone hovers at a height meters directly above a point on level ground

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

Problem

A drone hovers at a height

h(t)

meters directly above a point

A

on level ground. A camera is positioned at point

C

, which is 50 m horizontally from

A

. The camera tracks the drone by rotating so that the angle of elevation

\theta(t)

(measured in radians) satisfies

\tan \theta=\dfrac{h(t)}{50}

. At an instant when

\theta=\tfrac{\pi}{6}

, the drone is rising vertically at a rate of 3 m/s. (a) Express

h

as a function of

\theta

and identify the relationship between

\dfrac{dh}{dt}

and

\dfrac{d\theta}{dt}

by differentiating with respect to time. (b) Determine

\dfrac{d\theta}{dt}

when

\theta=\tfrac{\pi}{6}

and

\dfrac{dh}{dt}=3

. Give exact values and appropriate units. (c) Compute the instantaneous rate of change of

\sec \theta

with respect to time at the same moment. Include units in your answer. (d) Interpret the sign of

\dfrac{d\theta}{dt}

found in part (b) in the physical context of the situation.