Suppose a snowball melts in such a way that it maintains a spherical shape. If the radius is decreasing at a rate of 0.75 cm per hour when the radius is 10 cm, how fast is the volume of the snowball decreasing at that instant?

Question

Quizplus · Accepted Answer

The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$. Differentiating both sides with respect to time ($t$) gives $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$. Substituting $r = 10$ cm and $\frac{dr}{dt} = -0.75$ cm/hr (negative because the radius is decreasing), we get $\frac{dV}{dt} = 4\pi (10)^2 (-0.75) = -300\pi$ cm$^3$/hr. The negative sign indicates the volume is decreasing. The numerical value of $-300\pi$ is approximately $-942.5$ cm$^3$/hr, hence the correct answer is 942.5 cm$^3$/hr.

Suppose a Snowball Melts in Such a Way That It