Question

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. If the length of the cylinder is increasing by 0.4 cm per second, find the rate at which the radius is increasing when the radius is 2 cm and the length is 6 cm.

Answer 1

The only way I've done this is with calculus and derivatives.

Basically, you need a formula that dictates what is going on.
In this case that formula is Volume=Area*Length
However because we do not know the area but instead the radius, You need to put the formula in terms of the radius: Volume = π*(radius^2)*length

Then I would make a list of all the variable that you know:
(PS. I would first make sure that all the units in the problem are the same.  If not, adjust them so they are the same)
Volume: Find the volume at the set point in time by using this formula and plugging in the values of 2cm and 6cm for the radius and the length respectively.
This makes Volume=24π
It was also given that at the set time, the radius= 2 and length=6

However, we also need the rates of change for each of these variables.
A rate of change is a derivative so I will call them dv/dt, dr/dt. and dL/dt for the volume, radius, and length.
Since, we are finding dr/dt, we leave it as the variable dr/dt.
dL/dt was given as .4 (it is positive because it is increasing)
This is where you need common sense and a basic understanding of rate of change: dv/dt is 0 because even though it was not technically given, no clay is added or taken away so the volume must stay constant making how much it changes by equal to 0.

Now that we have the list, it is time to return to the formula.  First though we need to take the derivative of the formula.  Thus,
Volume = π*(radius^2)*length
becomes: 
dv/dt=(π*(radius^2)*dL/dt) + (2π*radius*length*dr/dt)
This was done with the product rule for derivative if you did not figure it out.

Then plug in the numbers from the list:
0=(π*4*.4)+(2π*2*6*dr/dt)
Then solve algebraically.
1.6π+24π*dr/dt=0
24π*dr/dt=-1.6π
dr/dt=-1/15

Then add units based on the problem:
-1/15 cm/second is the rate at which the radius is increasing (because it is negative, the radius is actually decreasing).