Question

A spherical snowball is melting in such a way that its radius is decreasing at a rate of 0.1 cm/min. At what rate is the volume of the snowball decreasing when the radius is 8 cm. (Note the answer is a positive number).

Answer 1

Answer: 80.384 cubic cm /min

Explanation:

Let V denote the volume and r denotes the radius of the spherical snowball .

Given : $\dfrac{dr}{dt}=-0.1\text{cm/min}$

We know that the volume of a sphere is given by :-

$V=\dfrac{4}{3}\pi r^3$

Differentiating on the both sides w.r.t. t (time) ,w e get

$\dfrac{dV}{dt}=\dfrac{4}{3}\pi(3r^2)\dfrac{dr}{dt}\\\\\Rightarrow\ \dfrac{dV}{dt}=4\pi r^2 (-0.1)=-0.4\pi r^2$

When r= 8 cm

$\dfrac{dV}{dt}=-0.4(3.14)(8)^2=-80.384$

Hence, the volume of the snowball decreasing at the rate of 80.384 cubic cm /min.