# The radius of a sphere is increasing at a rate of 5 mm/s. How fast is the volume increasing when...

## Question:

The radius of a sphere is increasing at a rate of 5 mm/s.

How fast is the volume increasing when the diameter is 40 mm?

Evaluate the answer numerically.

(Round the answer to the nearest whole number)

## Rate of Change:

We know that the volume of the sphere is a function of radius. The rate of change of radius is given. We will differentiate the volume function with respect to time using the chain rule of derivatives. We will then find the rate of change of volume.

## Answer and Explanation:

We know that the volume of the sphere formula is

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

We will differentiate with respect to time:

$$\frac{\mathrm{d} V}{\mathrm{d} t}=\frac{4}{3}\pi r^2\frac{\mathrm{d} r}{\mathrm{d} t}\\ $$

We know that:

$$\frac{\mathrm{d} r}{\mathrm{d} t}=5\\ r=\frac{d}{2}=20\\ \frac{\mathrm{d} V}{\mathrm{d} t}=\frac{8000\pi}{3} $$

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