Mass of the moon is 1/81 of the earth but gravitational pull is 1/6 of the earth . It is due to the fact that

Options

A. Radius of the moon is 81/6 of the earth
B. Radius of the Earth is `sqrt(81/6)` of the moon
C. Moon is the satellite of the earth
D. none

Answer

To solve the problem, we need to analyze the relationship between the gravitational pull of the Earth and the Moon, given their respective masses and the gravitational acceleration they produce.

### Step-by-Step Solution:

1. **Understanding Gravitational Pull**:
The gravitational pull (or acceleration due to gravity) on a celestial body is given by the formula:
\[ g = \frac{GM}{R^2} \]
where \( g \) is the acceleration due to gravity, \( G \) is the universal gravitational constant, \( M \) is the mass of the body, and \( R \) is its radius.

2. **Given Data**:
- Mass of the Moon (\( M_m \)) = \( \frac{1}{81} M_e \) (where \( M_e \) is the mass of the Earth)
- Gravitational pull of the Moon (\( g_m \)) = \( \frac{1}{6} g_e \) (where \( g_e \) is the gravitational pull of the Earth)

3. **Setting Up the Ratios**:
Using the formula for gravitational pull, we can express the gravitational accelerations for both the Earth and the Moon:
\[ g_e = \frac{GM_e}{R_e^2} \]
\[ g_m = \frac{GM_m}{R_m^2} = \frac{G \left(\frac{1}{81} M_e\right)}{R_m^2} \]

4. **Relating the Two Gravitational Pulls**:
Since we know \( g_m = \frac{1}{6} g_e \), we can write:
\[ \frac{G \left(\frac{1}{81} M_e\right)}{R_m^2} = \frac{1}{6} \left(\frac{GM_e}{R_e^2}\right) \]

5. **Simplifying the Equation**:
Canceling \( G \) and \( M_e \) from both sides, we get:
\[ \frac{1/81}{R_m^2} = \frac{1/6}{R_e^2} \]

6. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 6 \cdot \frac{1}{81} R_e^2 = R_m^2 \]
\[ R_m^2 = \frac{6}{81} R_e^2 \]

7. **Finding the Ratio of Radii**:
Taking the square root of both sides:
\[ R_m = R_e \cdot \sqrt{\frac{6}{81}} = R_e \cdot \frac{\sqrt{6}}{9} \]

8. **Finding the Ratio of Radii in Terms of Earth**:
Rearranging gives:
\[ \frac{R_e}{R_m} = \frac{9}{\sqrt{6}} \]

9. **Conclusion**:
The ratio of the radius of the Earth to the radius of the Moon is \( \frac{9}{\sqrt{6}} \), which corresponds to option 2 in the question.

### Final Answer:
The correct answer is option 2: the radius of the Earth is \( \sqrt{\frac{81}{6}} \) times that of the Moon.