The mean radius of a planet is `6.67xx10^(3)` km. The acceleration due to gravity on its surface is `10 m//s^(2)`. If `G = 6.67xx10^(-11)Nm^(2)//kg^(2)`, then the mass of the planet will be `[R=6.67xx10^(6)m]`

Options

A. `6.67xx10^(20)kg`
B. `6.67xx10^(24)kg`
C. `9.8xx10^(23)kg`
D. `13.34xx10^(23)kg`

Explanation

To find the mass of the planet, we can use the formula that relates the acceleration due to gravity (g) at the surface of a planet to its mass (M) and radius (R):

\[ g = \frac{GM}{R^2} \]

Where:
- \( g \) is the acceleration due to gravity (10 m/s²),
- \( G \) is the universal gravitational constant (\( 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)),
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step-by-Step Solution:

1. **Convert the radius from kilometers to meters**:
\[ R = 6.67 \times 10^3 \, \text{km} = 6.67 \times 10^3 \times 10^3 \, \text{m} = 6.67 \times 10^6 \, \text{m} \]

2. **Rearrange the formula to solve for mass (M)**:
\[ M = \frac{gR^2}{G} \]

3. **Substitute the known values into the equation**:
- \( g = 10 \, \text{m/s}^2 \)
- \( R = 6.67 \times 10^6 \, \text{m} \)
- \( G = 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)

\[ M = \frac{10 \times (6.67 \times 10^6)^2}{6.67 \times 10^{-11}} \]

4. **Calculate \( R^2 \)**:
\[ R^2 = (6.67 \times 10^6)^2 = 44.4889 \times 10^{12} \, \text{m}^2 \]

5. **Substitute \( R^2 \) back into the mass equation**:
\[ M = \frac{10 \times 44.4889 \times 10^{12}}{6.67 \times 10^{-11}} \]

6. **Perform the multiplication in the numerator**:
\[ M = \frac{444.889 \times 10^{12}}{6.67 \times 10^{-11}} \]

7. **Divide the numerator by the denominator**:
\[ M = 6.67 \times 10^{22} \, \text{kg} \]

### Final Answer:
The mass of the planet is approximately \( 6.67 \times 10^{22} \, \text{kg} \).