QPaperGen

The rate of change of the area of a circle with respect to its radius \(r\) at \(r = 6,\text{cm}\) is to be determined.\[\]

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

Differentiate with respect to \(r\):

\[\dfrac{dA}{dr} = \dfrac{d}{dr}(\pi r^2) = 2\pi r\]

Now, substitute \(r = 6\):

\[\dfrac{dA}{dr} = 2\pi \cdot 6 = 12\pi,\text{cm}^2/\text{cm}\]

Final Answer:
The rate of change of the area at \(r = 6,\text{cm}\) is \(12\pi,\text{cm}^2/\text{cm}\).