A spherical balloon has a variable radius. Find the rate at which its volume is increasing with respect to radius when $r = 10,\text{cm}$.

Q: A spherical balloon has a variable radius. Find the rate at which its volume is increasing with respect to radius when $r = 10,\text{cm}$.

Explanation: Volume of a sphere: $V = \frac{4}{3}\pi r^3$\[\] Differentiating with respect to $r$: \[\frac{dV}{dr} = \frac{d}{dr}\left(\frac{4}{3}\pi r^3\right) = 4\pi r^2\] When $r = 10,\text{cm}$: \[\frac{dV}{dr} = 4\pi (10)^2 = 400\pi \quad\text{cm}^3/\text{cm}\]

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Explanation: Volume of a sphere: $V = \frac{4}{3}\pi r^3$

Differentiating with respect to $r$:

$$\frac{dV}{dr} = \frac{d}{dr}\left(\frac{4}{3}\pi r^3\right) = 4\pi r^2$$

When $r = 10,\text{cm}$:

$$\frac{dV}{dr} = 4\pi (10)^2 = 400\pi \quad\text{cm}^3/\text{cm}$$

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