Answer to Question #104854 in Calculus for Kassim

Question

A spherical snowball is melting in such a way
that its diameter is decreasing at rate of 0.1 cm/min. At what
rate is the volume of the snowball decreasing when the diameter
is 13 cm. (Note the answer is a positive number)

Answer 1

Given

Diameter if sphere is decreasing at the rate of 0.1 cm/min

$i.e.\:\frac{dD}{dt}=-0.1\:\frac{cm}{min}$ (negative sign means diameter is decreasing)

We need to find rate of volume of snowball decreasing with respect to time $(\frac{dV}{dt})$ when diameter (D) is 13 cm.

we know volume of Sphere is

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

we also know radius is equal to half of diameter

Substituting $r=\frac{D}{2}$ in above equation

$V=\frac{4}{3}\pi \left(\frac{D}{2}\right)^3$

$V=\frac{\pi }{6} D^3$

Differentiating with respect to time

$\frac{dV}{dt}=\frac{\pi }{6}\cdot 3D^2\cdot \frac{dD}{dt}$

substituting values

$\frac{dV}{dt}=\frac{\pi }{6}\cdot 3\left(13\:cm\right)^2\cdot \left(-0.1\:\frac{cm}{min}\right)$

$\frac{dV}{dt}=-\frac{169\pi }{20}\:\frac{cm^3}{min}\approx -26.546\:\frac{cm^3}{min}$

Negative sign indicate volume is decreasing with respect to time.

Volume is decreasing at the rate of $\frac{169\pi }{20}$ $cm^3/min$ or 26.546 $cm^3/min$

Note:

1) Round answer as per requirements or write in terms of $\pi$

2) When we are mentioning "volume is decreasing" then we don't need to give negative sign and so it (answer) will be a positive number. But when we are writing in mathematical form $\frac{dV}{dt}$ , we need to give -ve sign to show that it is decreasing.