Question

A spherical snowball is melting in such a way that its radius is decreasing at a rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the radius is 17 cm. (Note the answer is a positive number).

Answer 1

Answer:

726.34 cm/min

Step-by-step explanation:

Volume of a sphere:

The volume of a sphere is given by the following equation:

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

In which r is the radius.

Implicit derivatives:

This question is solving by implicit derivatives. We derivate V and r, implicitly as function of t. So

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

A spherical snowball is melting in such a way that its radius is decreasing at a rate of 0.2 cm/min.

This means that $\frac{dr}{dt} = -0.2$

At what rate is the volume of the snowball decreasing when the radius is 17 cm.

This is $\frac{dV}{dt}$ when $r = 17$

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

$\frac{dV}{dt} = 4\pi*(17)^2*(-0.2)$

$\frac{dV}{dt} = -726.34$

This means that the volume of the snowball is decreasing at a rate of 726.34 cm/min when the radius is 17 cm.