The volume of a right circular cone. Whose radius of the base in one-third of its altitude, and the volume of a hemisphere are equal. The ratio of the radii of the cone and the hemisphere is

Options

A. `""^(3)sqrt(3): ""^(3) sqrt(2)`
B. `2:3`
C. `""^(3) sqrt(2):1`
D. `""^(3) sqrt(2): ""^(3) sqrt(3)`

Answer

To solve the problem, we need to find the ratio of the radii of a right circular cone and a hemisphere, given that the volume of the cone is equal to the volume of the hemisphere, and the radius of the cone is one-third of its altitude.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the radius of the cone be \( r_1 \).
- Let the height (altitude) of the cone be \( h \).
- Let the radius of the hemisphere be \( r_2 \).

2. **Relationship Between Radius and Height of the Cone:**
- According to the problem, the radius of the base of the cone is one-third of its altitude:
\[ r_1 = \frac{h}{3} \]
- From this, we can express the height in terms of the radius:
\[ h = 3r_1 \]

3. **Volume of the Cone:**
- The formula for the volume \( V \) of a right circular cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r_1^2 h \]
- Substituting \( h = 3r_1 \) into the volume formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r_1^2 (3r_1) = \pi r_1^3 \]

4. **Volume of the Hemisphere:**
- The formula for the volume \( V \) of a hemisphere is given by:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r_2^3 \]

5. **Setting the Volumes Equal:**
- Since the volumes of the cone and the hemisphere are equal:
\[ \pi r_1^3 = \frac{2}{3} \pi r_2^3 \]
- We can cancel \( \pi \) from both sides:
\[ r_1^3 = \frac{2}{3} r_2^3 \]

6. **Finding the Ratio of Radii:**
- Rearranging the equation gives:
\[ \frac{r_1^3}{r_2^3} = \frac{2}{3} \]
- Taking the cube root of both sides:
\[ \frac{r_1}{r_2} = \sqrt[3]{\frac{2}{3}} \]

7. **Final Ratio:**
- Thus, the ratio of the radii of the cone to the hemisphere is:
\[ r_1 : r_2 = \sqrt[3]{2} : \sqrt[3]{3} \]