A 15-foot ladder leans against a vertical wall on a level surface

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

Problem

A 15-foot ladder leans against a vertical wall on a level surface. At time

t=0

, when the foot of the ladder is 9 ft from the wall, the foot begins to slide directly away from the wall with an initial horizontal velocity of 3 ft/s and a constant horizontal acceleration of 0.20 ft/s

^{2}

. Throughout the motion, let

x(t)

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

y(t)

be the height (in feet) of the top of the ladder above the floor. (a) Express

x(t)

for

t \ge 0

assuming the motion begins as described. (b) Using an appropriate relationship between

x(t)

and

y(t)

, derive

y(t)

implicitly as a function of time. (c) Determine the instantaneous vertical velocity

\dfrac{dy}{dt}

of the top of the ladder at

t=0

. (d) Determine the instantaneous vertical acceleration

\dfrac{d^{2}y}{dt^{2}}

of the top of the ladder at

t=0

and interpret its sign in the context of the ladder’s motion.

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