A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere. What is the ratio of the height to the radius of the cone so formed?

Question

Inside a triangular park, a flower bed forms a triangle similar to the park's boundary. A uniform path runs around the flower bed, so that each side of the park is exactly double the corresponding side of the flower bed. What is the ratio of the area of the path to the area of the flower bed?

Answer 1

Since the sphere's volume formula and the cone's volume formula share a common $\pi r^2$ (or $\pi r^3$) factor, most of the algebra cancels away quickly.

Sphere volume: $\frac{4}{3}\pi r^3$.
Cone volume with the same radius $r$: $\frac{1}{3}\pi r^2 h$.
Setting them equal: $\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h$.
Dividing both sides by $\frac{1}{3}\pi r^2$: $4r = h$.

This gives a height-to-radius ratio of exactly $4:1$, which does not appear among the first three options.

So the correct answer is option D, none of these.

Answer 2

Since the sphere's volume formula and the cone's volume formula share a common $\pi r^2$ (or $\pi r^3$) factor, most of the algebra cancels away quickly.

Sphere volume: $\frac{4}{3}\pi r^3$.
Cone volume with the same radius $r$: $\frac{1}{3}\pi r^2 h$.
Setting them equal: $\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h$.
Dividing both sides by $\frac{1}{3}\pi r^2$: $4r = h$.

This gives a height-to-radius ratio of exactly $4:1$, which does not appear among the first three options.

So the correct answer is option D, none of these.

A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere. What is the ratio of the height to the radius of the cone so formed?

Show Hint

The Correct Option is D

Solution and Explanation

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