A spherical snowball is melting in such a way that its radius decreases at a rate of 0.05 centimeters per minute. Find the rate at which the volume of the snowball is decreasing when the radius is 10 centimeters.

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Topic: Calculus Posted 1 year ago

A spherical snowball is melting in such a way that its radius decreases at a rate of 0.05 centimeters per minute. Find the rate at which the volume of the snowball is decreasing when the radius is 10 centimeters.

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Koala Tutor

Nov 1, 2023

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We are given that $\frac{dr}{dt} = -0.05$ cm/min and we desire to find $\frac{dV}{dt}$ when $r = 10.$

The volume $V$ of a sphere with radius $r$ is given by $V = \frac{4}{3} \pi r^3.$ Taking the derivative with respect to $t,$ we get
\[\frac{dV}{dt} = 4 \pi r^2 \cdot \frac{dr}{dt}.\]Substituting the given information, we can compute
\[\left. \frac{dV}{dt} \right|_{r = 10} = 4 \pi (10^2)(-0.05) = \boxed{-20 \pi} \text{ cm}^3/\text{min}.\]

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