The volume of a right circular cone, whose radius of the base is the same as five-ninth of its altitude and the volume of a sphere are equal. The ratio of the radii of the cone to the sphere is:

Options

A. `root(2)(4): root(3)(3)`
B. `1:1`
C. `root(2)(60) : 3`
D. `root(3)(3) : root(3)(2)`

Answer

To solve the problem, we need to find the ratio of the radii of a right circular cone and a sphere, given that their volumes are equal. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the altitude (height) of the cone be \( H \).
- Let the radius of the base of the cone be \( R \).
- According to the problem, the radius of the cone is \( R = \frac{5}{9} H \).

2. **Express Height in Terms of Radius**:
- From the equation \( R = \frac{5}{9} H \), we can express \( H \) in terms of \( R \):
\[ H = \frac{9}{5} R \]

3. **Volume of the Cone**:
- The formula for the volume \( V \) of a right circular cone is:
\[ V_{cone} = \frac{1}{3} \pi R^2 H \]
- Substitute \( H \) with \( \frac{9}{5} R \):
\[ V_{cone} = \frac{1}{3} \pi R^2 \left(\frac{9}{5} R\right) = \frac{3 \pi}{5} R^3 \]

4. **Volume of the Sphere**:
- The formula for the volume \( V \) of a sphere is:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
- Here, \( r \) is the radius of the sphere.

5. **Set Volumes Equal**:
- According to the problem, the volumes of the cone and sphere are equal:
\[ \frac{3 \pi}{5} R^3 = \frac{4}{3} \pi r^3 \]

6. **Cancel \( \pi \) from Both Sides**:
- We can cancel \( \pi \) from both sides:
\[ \frac{3}{5} R^3 = \frac{4}{3} r^3 \]

7. **Cross-Multiply to Solve for \( R^3 \) and \( r^3 \)**:
- Cross-multiplying gives:
\[ 3 \cdot 3 R^3 = 4 \cdot 5 r^3 \]
\[ 9 R^3 = 20 r^3 \]

8. **Find the Ratio of \( R^3 \) to \( r^3 \)**:
- Rearranging gives:
\[ \frac{R^3}{r^3} = \frac{20}{9} \]

9. **Take the Cube Root**:
- To find the ratio of the radii \( R \) and \( r \):
\[ \frac{R}{r} = \sqrt[3]{\frac{20}{9}} = \frac{\sqrt[3]{20}}{\sqrt[3]{9}} \]

### Final Ratio:
The final ratio of the radii of the cone to the sphere is:
\[ \frac{R}{r} = \frac{\sqrt[3]{20}}{\sqrt[3]{9}} \]