Question

a spherical snowball is melting in such a way that its radius is decreasing at rate of 0.3 cm/min. at what rate is the volume of the snowball decreasing when the radius is 12 cm. (note the answer is a positive number).

hint: the volume of a sphere of radius r is ( v=\frac{4}{3}pi r^{3} )

Explanation:

Step1: Differentiate the volume formula

The volume of a sphere is \(V=\frac{4}{3}\pi r^{3}\). Differentiate \(V\) with respect to time \(t\) using the chain - rule. \(\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}\).
First, find \(\frac{dV}{dr}\): \(\frac{dV}{dr}=\frac{4}{3}\pi\cdot3r^{2}=4\pi r^{2}\).

Step2: Substitute the given values

We know that \(\frac{dr}{dt}=- 0.3\) (negative because the radius is decreasing) and \(r = 12\).
Substitute into \(\frac{dV}{dt}=4\pi r^{2}\cdot\frac{dr}{dt}\).
\(\frac{dV}{dt}=4\pi(12)^{2}\cdot(-0.3)\).
Calculate \(4\pi\times144\times(- 0.3)=-172.8\pi\). But the problem asks for a positive number (since it asks for the rate at which the volume is decreasing, and we can take the absolute value).

Answer:

\(172.8\pi\approx542.87\)