Question

The balloon is in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder. The balloon is being inflated at the rate of 261π cubic centimeters per minute. At the instant the radius of the cylinder is 3 cm, the volume of the balloon is 144π cubic cm and the radius of the cylinder is increasing at the rate of 2 cm/min.

At this instant, what is the height of the cylinder?

Answer 1

To find the height of the cylinder, use the formula for the volume of a cylinder and the given information. Differentiate the formula with respect to time to find the rate of change of the height. Integrate the rate of change of the height to find the height.

Step-by-step explanation:

To find the height of the cylinder, we need to use the formula for the volume of a cylinder and the given information. 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.

Given that the radius of the cylinder is increasing at a rate of 2 cm/min and the volume is increasing at a rate of 261π cm³/min, we can differentiate the formula with respect to time to find the rate at which the height is changing.

dV/dt = π(2rdr/dt + r²dh/dt) = 261π

At the instant the radius is 3 cm, the volume is 144π cm³. Substituting these values into the equation, we can solve for dh/dt, which will give us the rate of change of the height. Once we have the rate of change of the height, we can integrate it to find the height.