Question

The volume is 200cm^3

I am trying to work out the change in the height of a cylinder
the volume of the cylinder V = pi*r^2*h
the rate of change is
1/1000πcm/sec

the current radius is 45cm

radius is 45cm and the height is increasing at a rate of 1/1000πcm/sec.
I need to calculate the increase in size of the radius hence r

Answer 1

rate of change of what? is it \(r\) that is changing?

Answer 2

I guess the volume doesn't change wrt time. If so,
$$V=\pi r^2 h$$
differentiating wrt time - t
$$\frac{dV}{dt}=\pi \left(2r\frac{dr}{dt}\right)h+\pi r^2\frac{dh}{dt}\\
0=\pi r\left(2\frac{dr}{dt}h+r\frac{dh}{dt}\right)\\
\frac{dr}{dt}=-\frac{r}{2h}\frac{dh}{dt}$$

Answer 3

i updated my answer. Also when i add the values in I guess r = 45
and the dh/dt = 1/1000pi. But dont understand what 2h is. Could someone please point me in the right direction

Answer 4

Ok,
h is the height of the cylinder when the radius is r.
In the question the volume is given
thus,
$$V=\pi r^2 h \implies h=\frac{V}{\pi r^2}$$
substituting with the previous derivation,
$$\frac{dr}{dt}=-\frac{2r}{\frac{V}{\pi r^2}}\frac{dh}{dt}\\
\frac{dr}{dt}=-\frac{2\pi r^3}{V}\frac{dh}{dt}$$

now all the parameters are provided thus you can find the rate of change in radius

Answer 5

thanks I understand what todo now