Answer:
dV/dt = 150.79 m^3/s
Explanation:
In order to calculate the rate of change of the volume, you calculate the derivative, respect to the radius of the sphere, of the volume of the sphere, as follow:
[tex]\frac{dV}{dt}=\frac{d}{dt}(\frac{4}{3}\pi r^3)[/tex] (1)
r: radius of the sphere
You calculate the derivative of the equation (1):
[tex]\frac{dV}{dt}=\frac{d}{dt}(\frac{4}{3}\pi r^3)=3\frac{4}{3}\pi r^2\frac{dr}{dt}=4\pi r^2\frac{dr}{dt}\\\\\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}[/tex](2)
where dr/dt = 3m/s
You replace the values of dr/dt and r=2m in the equation (2):
[tex]\frac{dV}{dt}=4\pi (2m)^2(3\frac{m}{s})=150.79\frac{m^3}{s}[/tex]
The rate of change of the sphere, when it has a radius of 2m, is 150.79m^3/s
