Answer:
The volume is increasing at a rate of 25,133 mm/s.
Step-by-step explanation:
Diameter is 40mm
Radius is half the diameter, so [tex]r = \frac{40}{2} = 20[/tex]
How fast is the volume increasing when the diameter is 40mm?
We have to apply implicit differentiation, of V and r in function of t. So
[tex]V = \frac{4}{3} \pi r^3[/tex]
[tex]\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}[/tex]
The radius of a sphere is increasing at a rate of 5 mm/s.
This means that [tex]\frac{dr}{dt} = 5[/tex]
Then
[tex]\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}[/tex]
[tex]\frac{dV}{dt} = 4\pi (20)^2(5)[/tex]
[tex]\frac{dV}{dt} = 25133[/tex]
The volume is increasing at a rate of 25,133 mm/s.
