A stone is thrown into a pond, creating a circular wave whose radius increases at the rate of 1 foot per second. In square feet per second, how fast is the area of the circular ripple increasing 3 seconds after the stone hits the water?
A)$\pi$
B)2$\pi$
C)$\frac { \pi } { 3 }$
D)6$\pi$
E)3$\pi$
F) $\frac { \pi } { 6 }$
G)$\frac { \pi } { 12 }$
H) $\frac { \pi } { 2 }$

Question

A stone is thrown into a pond, creating a circular wave whose radius increases at the rate of 1 foot per second. In square feet per second, how fast is the area of the circular ripple increasing 3 seconds after the stone hits the water?
A)$\pi$
B)2$\pi$
C)$\frac { \pi } { 3 }$
D)6$\pi$
E)3$\pi$
F) $\frac { \pi } { 6 }$
G)$\frac { \pi } { 12 }$ 
H) $\frac { \pi } { 2 }$

Quizplus · Accepted Answer

The area of a circle is given by $A = \pi r^2$, where $r$ is the radius. The rate of change of the area with respect to time ($\frac{dA}{dt}$) can be found by differentiating both sides of the equation with respect to $t$, giving $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. Given $\frac{dr}{dt} = 1$ foot per second and $r = 3$ feet at the time of interest, substituting these values gives $\frac{dA}{dt} = 2\pi(3)(1) = 6\pi$ square feet per second.

A Stone Is Thrown into a Pond, Creating a Circular π\piπ

A Stone Is Thrown into a Pond, Creating a Circular $\pi$