A ladder rests against a vertical wall

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

Problem

10\text{-ft}

ladder rests against a vertical wall. At the instant when the foot of the ladder is

6\text{ ft}

from the wall, the foot is sliding away from the wall at a rate of

2\text{ ft/s}

. Let

x

be the horizontal distance (in feet) from the foot of the ladder to the wall, let

y

be the vertical height (in feet) of the top of the ladder above the ground, and let

t

be time in seconds. (a) Write an equation that relates

x

and

y

for the ladder at any time

t

. (b) Differentiate your equation from part (a) with respect to time to obtain a relationship between

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

. Then calculate

\dfrac{dy}{dt}

at the instant described in the problem. (c) Interpret the sign of

\dfrac{dy}{dt}

in the context of the motion of the ladder.