A 15-ft ladder rests against a vertical wall on a smooth horizontal surface

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A 15-ft ladder rests against a vertical wall on a smooth horizontal surface. Starting at

t=0

seconds, the horizontal position of the foot of the ladder, measured in feet from the base of the wall, is modeled by the function

x(t)=0.5t^{2}+2t.

Assume the ladder always remains in contact with the wall and the ground. (a) Derive an expression for the vertical speed

\dfrac{dy}{dt}

of the top of the ladder as a function of

t

. (b) Determine the exact time

t>0

(if any) at which the magnitude of

\dfrac{dy}{dt}

equals 3 ft/s. Justify that your result satisfies all physical constraints of the scenario. (c) Compute the vertical acceleration

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

of the top of the ladder at the instant when

x=6

ft, and interpret its sign in context.