# The radius r of a circle is increasing at a rate of 6 centimeters per minute. (a) Find the rate...

## Question:

The radius {eq}r {/eq} of a circle is increasing at a rate of 6 centimeters per minute.

(a) Find the rate of change of the area when {eq}r = 14 {/eq} centimeters.

(b) Find the rate of change of the area when {eq}r = 34 {/eq} centimeters.

## Rate of Change:

We have been given a circle whose radius increases with a constant rate. We have to find the rate of change of area with two given radii. We will apply the derivatives using the chain rule to the area function and get the rate of change of area in terms of area in terms of rate of change of radius.

## Answer and Explanation:

{eq}\frac{\mathrm{d} r}{\mathrm{d} t}=6~cm/min\\ A=\pi r^2\\ \text{Differentiating with respect to time:}\\ \frac{\mathrm{d} A}{\mathrm{d} t}=2\pi r\frac{\mathrm{d} r}{\mathrm{d} t}\\ \text{When :}\\ r=14\\ \frac{\mathrm{d} A}{\mathrm{d} t}=168 \pi~cm squared/min\\ \text{When:}\\ r=34\\ \frac{\mathrm{d} A}{\mathrm{d} t}=408 \pi ~cm squared/min\ {/eq}

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