Problem:
A right circular cone of volume A, a right circular cylinder of volume M, and a sphere of volume C all have the same radius, and the common height of the cone and the cylinder is equal to the diameter of the sphere. Then
Answer Choices:
A. A−M+C=0
B. A+M=C
C. 2A=M+C
D. A2−M2+C2=0
E. 2A+2M=3C
Solution:
Suppose the sphere has radius r. We can write the volumes of the three solids as functions of r as follows:
Volume of cone =A=31​πr2(2r)=32​πr3, Volume of cylinder =M=πr2(2r)=2πr3, and Volume of sphere =C=34​πr3.​
Thus, A−M+C=0.
Note: The AMC logo is designed to show this classical result of Archimedes.