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saveliy_v [14]
2 years ago
8

The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing when the diameter is 60 mm

Mathematics
1 answer:
mina [271]2 years ago
5 0

Answer:

The volume is increasing at a rate of 33929.3 cubic millimeters per second.

Step-by-step explanation:

Volume of a sphere:

The volume of a sphere of radius r is given by:

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

In this question:

We have to derivate V and r implicitly in function of time, so:

\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}

The radius of a sphere is increasing at a rate of 3 mm/s.

This means that \frac{dr}{dt} = 3

How fast is the volume increasing when the diameter is 60 mm?

Radius is half the diameter, so r = 30. We have to find \frac{dV}{dt}. So

\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}

\frac{dV}{dt} = 4\pi (30)^2(3) = 33929.3

The volume is increasing at a rate of 33929.3 cubic millimeters per second.

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