A searchlight is placed on level ground from the vertical face of a warehouse wall…

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

Problem

A searchlight is placed on level ground

60\,\text{m}

from the vertical face of a warehouse wall and is aimed horizontally at the wall. A worker who is

2.0\,\text{m}

tall walks briskly from the searchlight toward the wall at a constant speed of

1.5\,\text{m/s}

. The light source is located at ground level. For each time

t \ge 0

seconds, let •

x(t)

be the distance (in meters) between the worker and the searchlight, •

s(t)

be the height (in meters) of the top of the worker’s shadow on the wall. (a) Derive a relation between

x

and

s

. (b) Differentiate your relation implicitly with respect to time to obtain

\frac{ds}{dt}

in terms of

x

\frac{dx}{dt}

, and any needed constants. (c) Find the rate at which the top of the shadow is descending (

\frac{ds}{dt}

) at the moment the worker is

10\,\text{m}

from the wall. Include correct units. (d) Evaluate the limit of

\frac{ds}{dt}

as the worker approaches the wall and explain what this result indicates about the behavior of the shadow just before the worker reaches the wall.