A -ft ladder leans against a vertical wall

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

Problem

15

-ft ladder leans against a vertical wall. The bottom of the ladder slides away from the wall along level ground so that the horizontal distance

x(t)

(in feet) between the bottom of the ladder and the wall is increasing at a constant rate of

3\text{ ft/s}

. Let

y(t)

be the height of the top of the ladder above the ground at time

t

seconds, and let

A(t)=\tfrac12 x(t)\,y(t)

denote the area of the right triangle formed by the ladder, the wall, and the ground. (a) Express

y

as a function of

x

using the length of the ladder. (b) Derive a formula for

\dfrac{dA}{dt}

in terms of

x

only. (c) Determine the sign of

\dfrac{dA}{dt}

when

x=9\text{ ft}

. Interpret the result in the context of the triangle’s area. (d) Find the value of

x

for which

\dfrac{dA}{dt}=0

. Explain how the sign of

\dfrac{dA}{dt}

on either side of this value indicates whether the triangle’s area attains a maximum or a minimum there.