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 0.4 centimeters per second. What is the rate at which the radius is changing when the radius is 4 centimeters and the height is 7 centimeters?
To find the rate at which the radius is changing, we need to use the related rates problem solving technique.
Let's assign variables to the quantities involved:
- Let h be the height of the cylinder in centimeters.
- Let r be the radius of the cylinder in centimeters.
- Let dh/dt be the rate at which the height is changing, which is given as 0.4 cm/s.
We need to find dr/dt, the rate at which the radius is changing. We are given the values of r = 4 cm and h = 7 cm.
We know the formula for the volume of a cylinder is V = πr^2h. Since the clay is not being lost, the volume of the cylinder remains constant.
Differentiating both sides of the volume formula with respect to time t, we get:
dV/dt = d(πr^2h)/dt
Since dV/dt is zero (as the volume is not changing), we have:
0 = d(πr^2h)/dt
Next, we use the chain rule to differentiate the volume formula:
0 = d(πr^2h)/dt = π(2r)(dr/dt)h + πr^2(dh/dt)
Since we are looking for dr/dt, we rearrange the equation:
0 = 2πrh(dr/dt) + πr^2(dh/dt)
Now we substitute the given values:
- r = 4 cm
- h = 7 cm
- dh/dt = 0.4 cm/s
0 = 2π(4)(dr/dt) + π(4^2)(0.4)
0 = 8π(dr/dt) + 16π(0.4)
0 = 8π(dr/dt) + 6.4π
Now we isolate dr/dt:
-8π(dr/dt) = 6.4π
dr/dt = (6.4π) / (-8π)
dr/dt = -0.8 cm/s
Therefore, the rate at which the radius is changing when the radius is 4 cm and the height is 7 cm is -0.8 cm/s.