The radius of a circular puddle is growing at a rate of $8 \frac { \mathrm { cm } } { \mathrm { s } }$ . How fast is the area growing at the instant when it equals 20 cm² Round your answer to the nearest whole number. A) $127 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ B) $1005 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ C) $3 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ D) $72 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ E) $63 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$

Question

Quizplus · Accepted Answer

The area of a circle is given by $A = \pi r^2$. Differentiating both sides with respect to time $t$ gives $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. Given $\frac{dr}{dt} = 8 \frac{cm}{s}$ and $r = 20 cm$, substituting these values gives $\frac{dA}{dt} = 2\pi(20)(8) = 320\pi \approx 1005 \frac{cm^2}{s}$.

The Radius of a Circular Puddle Is Growing at a Rate