The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing (in mm^3/s) when the diameter is 60 mm

Answer:

The volume is increasing at a rate of 1508 cubic millimeters per second when the diameter is 60 mm.

Step-by-step explanation:

Volume of a sphere:

The volume of a sphere is given by the following equation:

[tex]V = \frac{4\pi r^3}{3}[/tex]

In which r is the radius.

Implicit derivatives:

This question is solving by implicit derivatives. We derivate V and r, implicitly as function of t. So

[tex]\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}[/tex]

The radius of a sphere is increasing at a rate of 4 mm/s.

This means that [tex]\frac{dr}{dt} = 4[/tex]

How fast is the volume increasing (in mm^3/s) when the diameter is 60 mm?

This is [tex]\frac{dV}{dt}[/tex] when [tex]r = \frac{d}{2} = \frac{60}{2} = 30[/tex]. So

[tex]\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}[/tex]

[tex]\frac{dV}{dt} = 4\pi*(30)^2*4[/tex]

[tex]\frac{dV}{dt} = 1508[/tex]

The volume is increasing at a rate of 1508 cubic millimeters per second when the diameter is 60 mm.

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