A 2-meter-tall person walks directly away from a 6-meter-tall lamppost at a constant…

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

Problem

A 2-meter-tall person walks directly away from a 6-meter-tall lamppost at a constant rate of 1 m/s. Let

s

denote the length of the person’s shadow at time

t

, and let

x

denote the person’s distance from the base of the lamppost at that same time. Which of the following statements about

\frac{ds}{dt}

is correct while the person is walking? (a)

\dfrac{ds}{dt}

equals

1\,\text{ m/s}

because the shadow lengthens at the same rate the person walks. (b)

\dfrac{ds}{dt}

is zero since the ratio

\dfrac{s}{x}

remains constant while the person walks. (c)

\dfrac{ds}{dt}

is positive because the shadow lengthens as the person moves farther from the lamppost. (d)

\dfrac{ds}{dt}

-\dfrac{1}{2}\,\text{ m/s}

, indicating the shadow shortens at half the walking speed.