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

726.34 cm/min

Explanation:

Volume of a sphere:

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

`V = (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

`(dV)/(dt) = 4\pi r^2 (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
`(dr)/(dt) = -0.2`

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

This is
`(dV)/(dt)` when
`r = 17`

`(dV)/(dt) = 4\pi r^2 (dr)/(dt)`

`(dV)/(dt) = 4\pi*(17)^2*(-0.2)`

`(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.