Multiple choice

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

  1. $A - M + C = 0$
  2. $A + M = C$
  3. $2A = M + C$
  4. $A^{2} - M^{2} + C^{2} = 0$
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A Correct answer
Explanation

Let the radius be r and the height be h=2r. The volume of the cone A is (1/3)pi*r^2*(2r) = (2/3)pi*r^3. The volume of the cylinder M is pi*r^2*(2r) = 2pi*r^3. The volume of the sphere C is (4/3)pi*r^3. Substituting these, A - M + C = (2/3)pi*r^3 - 2pi*r^3 + (4/3)pi*r^3 = (6/3)pi*r^3 - 2pi*r^3 = 0.

AI explanation

Let the common radius be r, so the common height h of the cone and cylinder equals 2r. The volumes are A = (1/3)πr^2(2r) = (2/3)πr^3, M = πr^2(2r) = 2πr^3, and C = (4/3)πr^3. Evaluating the expression A - M + C yields (2/3)πr^3 - 2πr^3 + (4/3)πr^3, which equals 0. This proves the relation A - M + C = 0.

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