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:
[tex]V = \frac{4\pi r^3}{3}[/tex]
In this question:
We have to derivate V and r implicitly in function of time, so:
[tex]\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}[/tex]
The radius of a sphere is increasing at a rate of 3 mm/s.
This means that [tex]\frac{dr}{dt} = 3[/tex]
How fast is the volume increasing when the diameter is 60 mm?
Radius is half the diameter, so [tex]r = 30[/tex]. We have to find [tex]\frac{dV}{dt}[/tex]. So
[tex]\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}[/tex]
[tex]\frac{dV}{dt} = 4\pi (30)^2(3) = 33929.3[/tex]
The volume is increasing at a rate of 33929.3 cubic millimeters per second.
