Related Rates: Spherical bubble. Air is pumped into a spherical bubble. When the bubble has a radius of 0.5 cm, the air is flowing into it at a rate of 0.2 cm³/sec. Find the rate at which the radius of the bubble is changing.

To solve this we need to relate the air flow to the radius. We can do this via the relationship between volume and radius and by using the data on air flow as the rate of change of the volume of the bubble:

.

Since , we can differentiate to get the related rates:

Now we solve for the rate of change of r:

Substitute the given values:

Note how the units cancel and the final result has units of length/time, compatible with .