Multiple choice

A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?

  1. $\sqrt {3} : 1$
  2. $1 : 1$
  3. $\dfrac {1}{2} : 1$
  4. $\sqrt {2} : 1$
  5. $2 : 1$
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B Correct answer
Explanation

For a cone inscribed in a hemisphere, the height of the cone is equal to the radius of the hemisphere because the vertex of the cone touches the center of the hemisphere's base plane if inverted, or the height is simply r. The ratio is 1:1.

AI explanation

When a right circular cone is inscribed in a hemisphere such that their bases coincide, the radius of the cone equals the radius of the hemisphere. Because the apex of the cone touches the top of the hemisphere's curved surface, the height of the cone exactly equals the common radius. Therefore, the ratio of the height of the cone to the radius of the hemisphere is 1 : 1.

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