Answer
Suppose the area of the circle
$$
A=\pi r^{2};
$$
The rate of change of the area of the circle when the radius is 7 ft. can be found by the following:
$$
\begin{aligned} \frac{d A}{d t} &=56 \pi.
\end{aligned}
$$
The rate of change of the area of the circle is $56 \pi$ ft. $^{2}$ /min.
Work Step by Step
Suppose the area of the circle
$$
A=\pi r^{2};
$$
the radius increasing at the rate of 4 ft per minute:=$\frac{dr}{dt}=4.$
The rate of change of the area of the circle when the radius is 7 ft. can be found by the following:
$$
\begin{aligned} \frac{d A}{d t} &=2 \pi r \frac{d r}{d t} \\ \frac{d A}{d t} &=2 \pi(7)(4) \\ \frac{d A}{d t} &=56 \pi.
\end{aligned}
$$
The rate of change of the area of the circle is $56 \pi$ ft. $^{2}$ /min.
