The radius of a circle is increasing at the rate of $0.7,\text{cm/s}$. What is the rate of increase of its circumference?

Q: The radius of a circle is increasing at the rate of $0.7,\text{cm/s}$. What is the rate of increase of its circumference?

Explanation: We know that $C = 2\pi r$. Differentiating with respect to time $t$: \[\frac{dC}{dt} = \frac{d}{dr}(2\pi r) \cdot \frac{dr}{dt} = 2\pi \cdot \frac{dr}{dt}\] Given $\frac{dr}{dt} = 0.7,\text{cm/s}$: \[\] $\frac{dC}{dt} = 2\pi (0.7) = 1.4\pi,\text{cm/s}$

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Explanation: We know that $C = 2\pi r$.
Differentiating with respect to time $t$:

$$\frac{dC}{dt} = \frac{d}{dr}(2\pi r) \cdot \frac{dr}{dt} = 2\pi \cdot \frac{dr}{dt}$$

Given $\frac{dr}{dt} = 0.7,\text{cm/s}$:

$\frac{dC}{dt} = 2\pi (0.7) = 1.4\pi,\text{cm/s}$

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