QPaperGen

We are asked to find how fast the area enclosed by the wave is increasing when the radius is \(8\,\text{cm}\).

We know that the area of a circle is given by \(A = \pi r^2\).

\[\]Differentiating both sides with respect to time \(t\):\[\]

\[\frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = \frac{d}{dr}(\pi r^2) \cdot \frac{dr}{dt} = 2\pi r \cdot \frac{dr}{dt}\]

Given \(\frac{dr}{dt} = 5\,\text{cm/s}\), we substitute the values:

\(\frac{dA}{dt} = 2\pi (8)(5) = 80\pi\,\text{cm}^2/\text{s}\)