The ratio of the weight of a body on the Earth’s surface to that on the surface of a planet is 9 : 4. The mass of the planet is `1/9` of that of the Earth. If ‘R’ is the radius of the Earth, then the radius of the planet is where n is ___________ . (Take the planets to have the same mass density)

Options

A. `R/4`
B. `R/9`
C. `R/3`
D. `R/2`

Explanation

To solve the problem, we need to find the radius of the planet in terms of the Earth's radius, given the ratio of weights and the mass relationship between the Earth and the planet.

### Step-by-Step Solution:

1. **Understanding the Weight Ratio**:
The weight of a body on the surface of a planet is given by the formula:
\[ W = \frac{GMm}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the body, and \( R \) is the radius of the planet.

2. **Setting Up the Weight Ratio**:
Let \( W_E \) be the weight on Earth and \( W_P \) be the weight on the planet. The problem states that:
\[ \frac{W_E}{W_P} = \frac{9}{4} \]

3. **Expressing Weights**:
For Earth:
\[ W_E = \frac{GM_E m}{R_E^2} \]
For the planet:
\[ W_P = \frac{GM_P m}{R_P^2} \]
where \( M_E \) is the mass of the Earth, \( R_E \) is the radius of the Earth, \( M_P \) is the mass of the planet, and \( R_P \) is the radius of the planet.

4. **Substituting Mass of the Planet**:
We know that \( M_P = \frac{1}{9} M_E \). Substituting this into the weight equation for the planet gives:
\[ W_P = \frac{G \left(\frac{1}{9} M_E\right) m}{R_P^2} = \frac{GM_E m}{9 R_P^2} \]

5. **Setting Up the Ratio**:
Now substituting \( W_E \) and \( W_P \) into the weight ratio:
\[ \frac{\frac{GM_E m}{R_E^2}}{\frac{GM_E m}{9 R_P^2}} = \frac{9}{4} \]
The \( GM_E m \) cancels out:
\[ \frac{R_P^2}{R_E^2} \cdot 9 = \frac{9}{4} \]

6. **Simplifying the Equation**:
Rearranging gives:
\[ R_P^2 = \frac{9}{4} \cdot \frac{R_E^2}{9} = \frac{R_E^2}{4} \]

7. **Finding the Radius of the Planet**:
Taking the square root of both sides:
\[ R_P = \frac{R_E}{2} \]

### Conclusion:
Thus, the radius of the planet \( R_P \) is \( \frac{R_E}{2} \).

### Final Answer:
Where \( n = 2 \).