Multiple choice

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is

  1. $\displaystyle\frac{1}{1}$
  2. $\displaystyle\frac{1}{2}$
  3. $\displaystyle\frac{2}{1}$
  4. $\displaystyle\frac{2}{3}$
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C Correct answer
Explanation

Let r be the radius of the sphere and the cone, and h be the height of the cone. The volume of the sphere is (4/3) * pi * r^3, and the volume of the cone is (1/3) * pi * r^2 * h. Given the cone volume is half the sphere volume, (1/3) * pi * r^2 * h = (1/2) * (4/3) * pi * r^3. Simplifying gives h = 2r, so the ratio h/r is 2/1.

AI explanation

The formula for the volume of the cone is 1/3 times pi times r squared times h, and the formula for the volume of the sphere is 4/3 times pi times r cubed. Setting the cone volume to one half of the sphere volume gives 1/3 times pi times r squared times h equals 1/2 times 4/3 times pi times r cubed. Solving for the height h yields 2r, so the ratio of the altitude of the cone to the radius of its base is 2 to 1.

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