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Question
Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 cm?
Solution
Let V be the volume of the spherical ball. Then,
V = \[\frac{4}{3}\pi r^3\]
\[\Rightarrow \frac{dV}{dr} = 4\pi r^2 \]
Thus, the rate of change of the volume of the sphere is \[4\pi r^2\].
\[\text { When r }= 2 cm, \]
\[ \left( \frac{dV}{dr} \right)_{r = 2} = 4\pi \left( 2 \right)^2_{} \]
\[ = 16\pi {cm}^3 /cm\]
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