A rectangle has one side of fixed length

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

Problem

10\text{ cm}

. The other side,

w(t)

, is increasing in such a way that

w(t)=3+\tfrac12 t^2,\qquad t\ge 0,

where

t

is measured in seconds and

w

in centimeters. Let

P(t)

denote the perimeter of the rectangle. (a) Write an explicit formula for

P(t)

in terms of

t

. Then find

\dfrac{dP}{dt}

. (b) Determine the exact time

t

at which the perimeter is increasing at a rate of

20\text{ cm/s}

. State the corresponding dimensions of the rectangle at that instant.