Respuesta :
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.
