Answer
a.
$A[r(t)]=\pi t^{4}$
represents the area of the oil slick as a function of time $t$
b.
$D_{t}A[r(t)]=4\pi t^{3}$
represents the rate of change of the spill area at time t.
At 100 minutes the area of the spill is changing at the rate of
$ 4,000,000\pi \ \ \mathrm{f}\mathrm{t}^{2}/ \min$
Work Step by Step
a.
$A[r(t)]$ represents the area of the oil slick as a function of time $t$
(t=the time in minutes passed after the beginning of the leak.)
$A(r)=\pi r^{2}$
In the definition of A(r), replace r with r(t)\begin{align*}
A[r(t)]&=\pi[r(t)]^{2} \quad & \\
&=\pi[t^{2}]^{2} \quad & \\
& =\pi t^{4}
\end{align*}
This function
b.
$D_{t}A[r(t)]$ represents the rate of change of the spill area at time t.
$D_{t}A[r(t)]=D_{t}[\pi t^{4}]=4\pi t^{3}$
For t=100,
$ D_{t}A[r(100)]=4\pi(100)^{3}=4,000,000\pi$
At 100 minutes the area of the spill is changing at the rate of
$ 4,000,000\pi \mathrm{f}\mathrm{t}^{2}/$min
