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geometry intermediate

Problem

A right circular cone is inscribed in a right circular cylinder. The volume of the cylinder is cubic centimeters. What is the number of cubic centimeters in the space inside the cylinder but outside the cone? Express your answer in terms of .

problem
Solution
A cylinder with radius and height has volume ; a cone with the same height and radius has volume . Thus we see the cone has the volume of the cylinder, so the space between the cylinder and cone has the volume of the cylinder, which is .
Final answer
48\pi