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 0.9 centimeters per second. What is the rate at which the radius is changing when the radius is 6 centimeters and the height is 9 centimeters?

Answer 1

The rate at which the radius is changing when the radius is 6 cm and the height is 9 cm is -1.35 cm/s.

Step-by-step explanation:

Since the volume of a cylinder is given by V = πr2h, we can use the formula to determine the rate at which the radius is changing.

Given: dh/dt = 0.9 cm/s, r = 6 cm, h = 9 cm

We need to find dr/dt when r = 6 cm and h = 9 cm.

To solve this problem, we use the chain rule of differentiation:

1. Differentiate both sides of the formula V = πr2h with respect to t to get dV/dt = 2πrh(dr/dt) + πr2(dh/dt).
2. Substitute the given values into the equation and solve for dr/dt.

By substituting the given values, we have dV/dt = 2π(6)(9)(dr/dt) + π(6)2(0.9).

Simplifying the equation, we find that dr/dt = -1.35 cm/s.