A rectangle has constant area

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

Problem

A=18\text{ m}^2

. At time

t

seconds, the length

L(t)

and width

W(t)

are differentiable functions with

L(t)W(t)=18

. (1) Derive an equation that relates

\dfrac{dL}{dt}

and

\dfrac{dW}{dt}

at any time

t

. Express

\dfrac{dW}{dt}

in terms of

L

W

, and

\dfrac{dL}{dt}

. Show all steps in your reasoning. (2) At the instant when

L=6\text{ m}

and

\dfrac{dL}{dt}=0.20\text{ m/s}

, determine the numerical value (including sign) of

\dfrac{dW}{dt}

. Interpret the sign of

\dfrac{dW}{dt}

in the context of the rectangle’s dimensions. (3) Explain how the relative magnitudes of

L

and

W

influence the magnitude of

\left|\dfrac{dW}{dt}\right|

when

\dfrac{dL}{dt}

is positive and fixed. Support your explanation with the relationship you derived in part (a).