Question

The height of a cylinder with a fixed radius of 4 cm is increasing at the rate of 2 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 14cm.

Answer 1

what is the formula for the volume of a cylinder ?

Answer 2

oh i think it is \\[V=\pi r^2h\] which in your case is \[V=16\pi h\] is that right?

Answer 3

if so that means
\[V'=16\pi h'\] and you are told \(h'=4\)
plug it in, see what you get

Answer 4

from your data, If I call h the height of the cylinder, and \[\alpha \]=2 cm/min the rate of increasing for h, we can write:
\[h(t)=h _{0}+\alpha t\]
where \[h _{0}\] is the initial height of cylinder, so if I call V the volume of cylinder, we can write:
\[V(t)=16\pi h(t)=16 \pi (h _{0}+a t)\]
from which, by derivation, I get:
\[\frac{ dV }{ dt }=16 \pi \ \alpha \]
I think it is the rate of increasing of the volume of the cylinder,
which is independent of t