(a) If $A$ is the area of a circle with radius $r$ and the circle expands as time passes, find $\frac{d A}{d t}$ in terms of $\frac{d r}{d t}$.
\[
\frac{d A}{d t}=\left(\square \frac{d r}{d t}\right.
\]
(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of $2 \mathrm{~m} / \mathrm{s}$, exactly how fast (in $\mathrm{m}^{2} / \mathrm{s}$ ) is the area of the spill increasing when the radius is $33 \mathrm{~m}$ ?

Step 1 ：The area of a circle is given by the formula \(A = \pi r^2\). We are asked to find \(\frac{d A}{d t}\), the rate of change of the area with respect to time.

Step 2 ：Taking the derivative of both sides of the area formula with respect to time \(t\), we get \(\frac{d A}{d t} = 2\pi r \frac{d r}{d t}\).

Step 3 ：So, \(\frac{d A}{d t} = 2\pi r \frac{d r}{d t}\) is the rate of change of the area of the circle with respect to time in terms of \(\frac{d r}{d t}\).

Step 4 ：For the second part of the question, we are given that the radius of the oil spill increases at a constant rate of \(2 \mathrm{~m} / \mathrm{s}\), so \(\frac{d r}{d t} = 2\).

Step 5 ：We are asked to find how fast the area of the spill is increasing when the radius is \(33 \mathrm{~m}\). We can substitute these values into the formula we derived earlier.

Step 6 ：Substituting \(r = 33\) and \(\frac{d r}{d t} = 2\) into the formula, we get \(\frac{d A}{d t} = 2\pi (33) (2)\).

Step 7 ：Solving this, we get \(\frac{d A}{d t} = 132\pi \mathrm{~m}^{2} / \mathrm{s}\).

Step 8 ：So, the area of the oil spill is increasing at a rate of \(132\pi \mathrm{~m}^{2} / \mathrm{s}\) when the radius is \(33 \mathrm{~m}\).

Step 9 ：Finally, we check our answer. The units are correct (\(\mathrm{m}^{2} / \mathrm{s}\)), and the rate of change of the area is positive, which makes sense because the oil spill is expanding. Therefore, our answer is reasonable.

Step 10 ：Thus, the rate of change of the area of the oil spill with respect to time when the radius is \(33 \mathrm{~m}\) is \(\boxed{132\pi \mathrm{~m}^{2} / \mathrm{s}}\).