The circumference of a circle is observed to be increasing uniformly at

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

Problem

The circumference

C

of a circle is observed to be increasing uniformly at

6\,\mathrm{cm\,s^{-1}}

. Determine how fast the area

A

of the circle is increasing at the instant when the radius is

14\,\mathrm{cm}

. (i) Express

C

in terms of the radius

r

. (ii) Differentiate your relation from part (i) with respect to time to find a formula for

\dfrac{dC}{dt}

in terms of

r

and

\dfrac{dr}{dt}

. (iii) Using the given rate of change of circumference, calculate

\dfrac{dr}{dt}

when

r=14\,\mathrm{cm}

. (iv) Finally, determine

\dfrac{dA}{dt}

at this instant and state the result with correct units.