A closed circular cylinder has height \(16\) in. and radius \(r\) in. The total surface area is \(A\) sq. in.

Prove that \[\begin{equation*} \frac{dA}{dr} = 4\pi(r+8). \end{equation*}\]

Use this result to calculate an approximation for the increase in area when the radius increases from \(4\) to \(4.02\) in., the height remaining constant. You may leave the answer in terms of \(\pi\).