1. ## Calc related rates

A potter forms a piece of clay into a cylinder. As he rolls it, the length, L, of the cylinder increases and the radius, R, decreases. Assume that no clay is lost in the process. Suppose the length of the cylinder is increasing by 0.2 cm per second.

What is the rate at which the volume is changing?

What is the rate at which the radius is changing when the radius is 2 cm and the length is 7 cm?

2. Originally Posted by bluejewballs
A potter forms a piece of clay into a cylinder. As he rolls it, the length, L, of the cylinder increases and the radius, R, decreases. Assume that no clay is lost in the process. Suppose the length of the cylinder is increasing by 0.2 cm per second.

What is the rate at which the volume is changing?

What is the rate at which the radius is changing when the radius is 2 cm and the length is 7 cm?
The Volume of a circular cylinder is $\displaystyle V=\pi r^2l$

so taking the derivative with respect to t we get

$\displaystyle \frac{dV}{dt}=\pi \underbrace{\left[ 2rl\frac{dr}{dt}+r^2\frac{dl}{dt}\right]}_{product Rule}$

Since we are told that "no clay is lost" the volume is constant and therefore $\displaystyle \frac{dV}{dt}=0$

for the last question

$\displaystyle 0=\pi \left[ 2(2cm)(7cm)\frac{.2cm}{s}+(2cm)^2\frac{dl}{dt}\rig ht] \iff \frac{dl}{dt} = -\frac{1.4cm}{s}$