Question

A hemisphere of radius 2 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.

Answer 1

The radius and height of the cylinder of maximum volume are both 2 units.

Step-by-step explanation:

To find the radius and height of the cylinder of maximum volume, we start by considering that the volume of the cylinder is given by the formula V = πr²h. Since the cylinder's axis is vertical and the top rim touches the hemisphere, the radius of the cylinder will be the same as the radius of the hemisphere, which is 2. We can also determine the height of the cylinder by considering that the top circular rim of the cylinder touches the hemisphere. Since the radius of the hemisphere is 2, the height of the cylinder will be equal to the remaining height of the hemisphere, which is 2. Therefore, the radius and height of the cylinder of maximum volume are both 2 units.