An observer stands 40 m from the base of a weather balloon that is rising vertically

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

An observer stands 40 m from the base of a weather balloon that is rising vertically. The angle of elevation

\theta

(in radians) between the observer’s line of sight and the ground satisfies

\theta = \arctan\!\left(\dfrac{h}{40}\right)

, where

h

meters is the balloon’s height. The balloon’s height is increasing at a constant rate. Which of the following statements correctly describes

\dfrac{d\theta}{dt}

at the instant shown? (a)

\dfrac{d\theta}{dt}

is negative because

\arctan

is a decreasing function for positive arguments. (b)

\dfrac{d\theta}{dt}

is zero since the horizontal distance from the observer to the balloon's base does not change. (c)

\dfrac{d\theta}{dt}

is positive because an increasing height increases

\theta

. (Correct) (d)

\dfrac{d\theta}{dt}

is negative because, as the balloon moves toward being overhead, the angle measured from the horizontal decreases.