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 18 cm. (Note the answer is a positive number). min 4 Hint: The volume of a sphere of radius ris V Question Help: D Video

Answer 1

Sol:

The decreasing rate of the radius is given by:

1)

dr dt 0.1 cm/min

The volume of the spherical snowball is given by:

(V

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

The decreasing rate of the volume is given by:

(V

$\frac{dV}{dt}=\frac{d}{dt}\left [\frac{4}{3}\pi r^3 \right ]$

$\Rightarrow \frac{dV}{dt}=\frac{4}{3}\pi\cdot 3\ r^2 \ \frac{dr}{dt}$

$\Rightarrow \frac{dV}{dt}=4\pi r^2\ \frac{dr}{dt}$

For , the rate of decreasing of the volume is:

$\frac{dV}{dt}=4\pi \cdot 18^2\cdot (0.1)\ \ \ \frac {cm^3}{min}$

$\Rightarrow \frac{dV}{dt}= 129.6\pi\ \ \frac {cm^3}{min}$

$\Rightarrow \frac{dV}{dt}\approx 407.15\ \ \frac {cm^3}{min}$

So, the answer is approximately

$407.15\ \ \frac {cm^3}{min}$