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Suppose that the radius \(r\) and area \(A=\pi r^2\) of a circle are differentiable functions of \(t\). Write an equation that relates \(\frac{dA}{dt}\) to \(\frac{dr}{dt}\).

1. \(2r\,\frac{dr}{dt}\)
2. \(2\pi r\,\frac{dr}{dt}\)
3. \(4\pi r\,\frac{dr}{dt}\)
4. \(3\pi r\,\frac{dr}{dt}\)

Correct answer: \(2\pi r\,\frac{dr}{dt}\)

Solution

Since \(A=\pi r^2\), differentiating with respect to \(t\) gives \(\frac{dA}{dt}=\pi\cdot 2r\frac{dr}{dt}=2\pi r\frac{dr}{dt}\). This is a direct application of the chain rule.

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