Question

The radius of a cylinder is increasing at a constant rate of 3 meters per second, and the volume is increasing at a rate of 108 cubic meters per second. At the instant when the height of the cylinder is 6 meters and the volume is 33 cubic meters, what is the rate of change of the height? The volume of a cylinder can be found with the equation V=πr²h. Round your answers to three decimal places.

Answer 1

The rate of change of the height of the cylinder is 1.203 m/s.

Step-by-step explanation:

To find the rate of change of the height of the cylinder, we need to use the relationship between the volume of a cylinder and its height.

The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.

We are given that the radius is increasing at a constant rate of 3 meters per second, and the volume is increasing at a rate of 108 cubic meters per second. At the instant when the height of the cylinder is 6 meters and the volume is 33 cubic meters, we need to find the rate of change of the height.

Since we know the formula for the volume, we can differentiate it with respect to time:

dV/dt = πr²(dh/dt)

Given that dV/dt = 108 and r = 6 (since the radius is increasing at a rate of 3 m/s, and at the instant the height is 6 m, the radius is half the height), we can substitute these values into the equation:

108 = π(6)²(dh/dt)

Now we can solve for dh/dt:

dh/dt = 108 / (π(6)²) = 1.203 m/s (rounded to three decimal places)