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 0.7 centimeters per second. what is the rate at which the radius is changing when the radius is 3 centimeters and the height is 9 centimeters?

Answer 1

To find the rate at which the radius is changing when the height is increasing, we can use related rates and the volume equation of a cylinder. By differentiating the volume equation and substituting the given values, we can find the rate at which the radius is changing.

Step-by-step explanation:

To find the rate at which the radius is changing, we can use related rates. We know that the volume of the cylinder remains constant, so we can differentiate the volume equation with respect to time to find an equation that relates the rates of change of height and radius.

V = πr^2h

0 = π(2rh · dr/dt + r^2 · dh/dt)

Solving for dr/dt, we get:

dr/dt = -(r/h) · dh/dt

Substituting the given values, when r = 3 cm and h = 9 cm, dh/dt = 0.7 cm/s, we can find dr/dt using the formula:

dr/dt = -(3/9) · 0.7 = -0.233 cm/s