Final answer:
The dimensions of the cylinder are a height of 12 feet and a diameter of 12 feet. The maximum volume of this cylinder is 432π cubic feet.
Step-by-step explanation:
To find the dimensions of the cylinder with the greatest volume that can be inscribed in the sphere, we need to relate the dimensions of the cylinder to the radius of the sphere. The cylinder will have a height equal to double the radius of the sphere, and the diameter of the cylinder will be equal to the diameter of the sphere. Therefore, the dimensions of the cylinder will be: height = 12 feet and diameter = 12 feet.
In order to calculate the maximum volume of this cylinder, we can use the formula for the volume of a cylinder: V = πr²h, where r is the radius of the cylinder and h is the height. Plugging in the values, we get: V = π(6 feet)²(12 feet) = 432π cubic feet. This is the maximum volume of the cylinder.