# If the radius of a sphere is increasing at a rate of 4 cm per second, how fast is the volume increasing when the diameter is 80 cm?

##### 1 Answer

12,800cm3s

#### Explanation:

This is a classic Related Rates problems. The idea behind Related Rates is that you have a geometric model that doesn't change, even as the numbers do change.

For example, this shape will remain a sphere even as it changes size. The relationship between a where's volume and it's radius is

As long as this *geometric relationship* doesn't change as the sphere grows, then we can derive this relationship implicitly, and find a new relationship between the rates of change.

Implicit differentiation is where we derive every variable in the formula, and in this case, we derive the formula with respect to time.

So we take the derivative of our sphere:

We were actually given

We are interested in the moment when the *diameter* is 80 cm, which is when the *radius* will be 40 cm.

The rate of increase of the volume is

And the units even work out correctly, since we should get a volume divided by time.

Hope this helps.