Average distance of the earth from the sun is `L_(1)`. If one year of the earth =D days, one year of another planet whose average distance from the sun is `L_(2)` will be

Options

A. `D((L_(2))/(L_(1)))^(1//2)` days
B. `D((L_(2))/(L_(1)))^(3//2)` days
C. `D((L_(2))/(L_(1)))^(2//3)` days
D. `D((L_(2))/(L_(1)))` days

Explanation

To solve the problem, we will use Kepler's Third Law of planetary motion, which relates the time period of a planet's orbit to its average distance from the Sun.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Average distance of Earth from the Sun: \( L_1 \)
- Time period of Earth (1 year): \( T_E = d \)
- Average distance of another planet from the Sun: \( L_2 \)
- Time period of the other planet: \( T_P \) (this is what we need to find).

2. **Apply Kepler's Third Law:**
According to Kepler's Third Law, the square of the time period of a planet is directly proportional to the cube of its average distance from the Sun. Mathematically, this can be expressed as:
\[ T^2 \propto R^3 \]
This means:
\[ T^2 = k R^3 \]
where \( k \) is a constant.

3. **Write the Equations for Earth and the Other Planet:**
For Earth:
\[ T_E^2 = k L_1^3 \]
For the other planet:
\[ T_P^2 = k L_2^3 \]

4. **Take the Ratio of the Two Equations:**
Dividing the equation for the other planet by the equation for Earth gives:
\[ \frac{T_P^2}{T_E^2} = \frac{L_2^3}{L_1^3} \]

5. **Substituting the Known Time Period for Earth:**
Since \( T_E = d \), we can substitute this into the equation:
\[ \frac{T_P^2}{d^2} = \frac{L_2^3}{L_1^3} \]

6. **Rearranging to Solve for \( T_P \):**
Multiplying both sides by \( d^2 \):
\[ T_P^2 = d^2 \cdot \frac{L_2^3}{L_1^3} \]
Taking the square root of both sides:
\[ T_P = d \cdot \left(\frac{L_2}{L_1}\right)^{3/2} \]

7. **Final Result:**
Therefore, the time period of the other planet is:
\[ T_P = d \cdot \left(\frac{L_2}{L_1}\right)^{3/2} \]