QPaperGen

The area of a circle, denoted by \(A\), with a radius \(r\), is given by the formula:\[A = \pi r^2\]To find the rate of change of the area with respect to its radius, we differentiate the area formula with respect to \(r\):\[\frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r\]Now, let's calculate this rate of change for specific values of the radius:

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1. When \(r = 3\) cm: Substitute \(r=3\) into the derivative: \[\frac{dA}{dr} = 2\pi (3) = 6\pi\] Therefore, when the radius is 3 cm, the area of the circle is changing at a rate of \(6\pi\) cm\(^2\)/s.
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2. When \(r = 4\) cm: Substitute \(r=4\) into the derivative: \[\frac{dA}{dr} = 2\pi (4) = 8\pi\] Therefore, when the radius is 4 cm, the area of the circle is changing at a rate of \(8\pi\) cm\(^2\)/s.