Question

The height of a cylinder with a fixed radius of 10 cm is increasing at the rate of 0.5 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 30cm.

Answer 1

dh/dt is given ...
find dv/dt ... dv/dt = pi * r^2 *dh/dt
r=10 cm ; dh/dt=o.5 cm/min

Answer 2

you should start by writing down the formula for the volume of a cylinder

Answer 3

\[ v= \pi r^2 h \]
fixed radius of 10 cm means r is constant.
height increases and volume increases. those variables are not constant.
take the derivative with respect to time
\[ \frac{d}{dt} v = \pi r^2\frac{d}{dt} h \]
we treat pi and r as constants

Answer 4

or simply
\[ \frac{dv}{dt} = \pi r^2\frac{dh}{dt} \]
or
\[ \dot{v} = \pi r^2 \dot{h}\]
(in physics, the dot over the variable is short-hand for derivative with respect to time. It was the notation Newton used)

now replace the variables with the given values