The area of a circular sun spot is growing at a rate of $600 \frac { \mathrm { km } ^ { 2 } } { \mathrm {~s} }$ . How fast is the radius growing at the instant when it equals 5,000 km Round your answer to three decimal places.
A) $\frac { \mathrm { km } } { \mathrm { s } }$
B) $0.019 \frac { \mathrm { km } } { \mathrm { s } }$
C) $0.038 \frac { \mathrm { km } } { \mathrm { s } }$
D) $8.333 \frac { \mathrm { km } } { \mathrm { s } }$
E) $0.060 \frac { \mathrm { km } } { \mathrm { s } }$

Question

The area of a circular sun spot is growing at a rate of $600 \frac { \mathrm { km } ^ { 2 } } { \mathrm {~s} }$ . How fast is the radius growing at the instant when it equals 5,000 km  Round your answer to three decimal places.  
A) $\frac { \mathrm { km } } { \mathrm { s } }$
B) $0.019 \frac { \mathrm { km } } { \mathrm { s } }$
C) $0.038 \frac { \mathrm { km } } { \mathrm { s } }$
D) $8.333 \frac { \mathrm { km } } { \mathrm { s } }$
E) $0.060 \frac { \mathrm { km } } { \mathrm { s } }$

Quizplus · Accepted Answer

Given the area of a circle $A = \pi r^2$, the rate of change of the area with respect to time ($\frac{dA}{dt}$) is $600 \frac{km^2}{s}$. To find the rate of change of the radius ($\frac{dr}{dt}$) when $r = 5000$ km, we differentiate $A = \pi r^2$ with respect to $t$, giving $2\pi r \frac{dr}{dt}$. Substituting $\frac{dA}{dt} = 600$ and $r = 5000$, we solve for $\frac{dr}{dt}$ to find the rate at which the radius is growing.

The Area of a Circular Sun Spot Is Growing at a Rate