The respective height and volume of a hemisphere and a right circular cylinder are equal, then the ratio of their radii is

Options

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

Answer

To solve the problem, we need to find the ratio of the radii of a hemisphere and a right circular cylinder given that their respective height and volume are equal.

### Step-by-Step Solution:

1. **Understand the Problem**:
- Let the height of the hemisphere be \( h \) and the radius be \( r_h \).
- Let the height of the cylinder be \( h_c \) and the radius be \( r_c \).
- According to the problem, the height and volume of both shapes are equal.

2. **Set the Heights Equal**:
- Since the height of the hemisphere is equal to the height of the cylinder, we can denote both heights as \( h = x \).

3. **Volume of the Hemisphere**:
- The volume \( V_h \) of a hemisphere is given by the formula:
\[ V_h = \frac{2}{3} \pi r_h^3 \]
- Since the radius of the hemisphere is also \( r_h = x \), we can substitute:
\[ V_h = \frac{2}{3} \pi x^3 \]

4. **Volume of the Cylinder**:
- The volume \( V_c \) of a right circular cylinder is given by the formula:
\[ V_c = \pi r_c^2 h_c \]
- Since the height of the cylinder is also \( h_c = x \), we can substitute:
\[ V_c = \pi r_c^2 x \]

5. **Set the Volumes Equal**:
- According to the problem, the volumes are equal:
\[ \frac{2}{3} \pi x^3 = \pi r_c^2 x \]
- We can cancel \( \pi \) and \( x \) (assuming \( x \neq 0 \)):
\[ \frac{2}{3} x^2 = r_c^2 \]

6. **Solve for the Radius of the Cylinder**:
- Rearranging gives:
\[ r_c^2 = \frac{2}{3} x^2 \]
- Taking the square root of both sides:
\[ r_c = x \sqrt{\frac{2}{3}} = x \frac{\sqrt{2}}{\sqrt{3}} \]

7. **Find the Ratio of the Radii**:
- The ratio of the radius of the hemisphere \( r_h \) to the radius of the cylinder \( r_c \) is:
\[ \frac{r_h}{r_c} = \frac{x}{x \frac{\sqrt{2}}{\sqrt{3}}} = \frac{\sqrt{3}}{\sqrt{2}} \]

### Final Result:
The ratio of their radii is:
\[ \frac{r_h}{r_c} = \frac{\sqrt{3}}{\sqrt{2}} \]