Related Rates problem: I'm not sure how to solve this problem. My answer came out as 0.05 cm/s, but it is incorrect...

Question

A potter forms a piece of clay into a right circular cylinder. As she rolls it, the height ℎ of the cylinder increases and the radius 'r' decreases. Assume that no clay is lost in the process. Suppose the height of the cylinder is increasing by 0.1 centimeters per second. What is the rate at which the radius is changing when the radius is 5 centimeters and the height is 6 centimeters?

Answer 1

Let

h(t) be the height at a time t,

r(t) be the radius at a time t.

Then the volume at a time t is

V(t) = πr²(t)h(t) (1)

For simplity of notation I will drop the argument (t).

Also I will write

h' = dh/dt

r' = dr/dt

V' = dV/dt

So (1) becomes

V = πr²h

Differentiate with respect to time

V' = π(2rr'h + r²h')

"No clay lost" means V' = 0.

So

0 = π(2rr'h + r²h')

Simplify

0 = 2r'h + rh'

Express

r' = -rh'/2h

Substitute numbers

r' = -5×0.1/2×6 ≈ -0.042 cm/s