The gravitational force between two bodies is decreased by `36%` when the distance between them is increased by `3m`. The initial distance between them is

Options

A. `6 m`
B. `9m`
C. `12 m`
D. `15 m`

Explanation

To solve the problem, we need to find the initial distance between two bodies given that the gravitational force between them decreases by 36% when the distance is increased by 3 meters.

### Step-by-Step Solution:

1. **Understanding the Gravitational Force Formula**:
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( d \) is given by:
\[ F = \frac{G m_1 m_2}{d^2} \]
where \( G \) is the gravitational constant.

2. **Setting Up the Problem**:
Let the initial distance between the two bodies be \( d \). When the distance is increased by 3 meters, the new distance becomes \( d + 3 \). The new gravitational force \( F' \) can be expressed as:
\[ F' = \frac{G m_1 m_2}{(d + 3)^2} \]

3. **Expressing the Decrease in Force**:
According to the problem, the gravitational force decreases by 36%. Therefore, the new force \( F' \) is:
\[ F' = F - 0.36F = 0.64F \]
Substituting the expression for \( F \):
\[ \frac{G m_1 m_2}{(d + 3)^2} = 0.64 \cdot \frac{G m_1 m_2}{d^2} \]

4. **Cancelling Common Terms**:
Since \( G m_1 m_2 \) is common on both sides, we can cancel it out:
\[ \frac{1}{(d + 3)^2} = 0.64 \cdot \frac{1}{d^2} \]

5. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ d^2 = 0.64 (d + 3)^2 \]

6. **Expanding the Equation**:
Expanding the right side:
\[ d^2 = 0.64 (d^2 + 6d + 9) \]
This simplifies to:
\[ d^2 = 0.64d^2 + 3.84d + 5.76 \]

7. **Rearranging the Equation**:
Bringing all terms to one side:
\[ d^2 - 0.64d^2 - 3.84d - 5.76 = 0 \]
This simplifies to:
\[ 0.36d^2 - 3.84d - 5.76 = 0 \]

8. **Multiplying Through by 100**:
To eliminate decimals, multiply the entire equation by 100:
\[ 36d^2 - 384d - 576 = 0 \]

9. **Using the Quadratic Formula**:
We can apply the quadratic formula \( d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 36 \), \( b = -384 \), and \( c = -576 \):
\[ d = \frac{384 \pm \sqrt{(-384)^2 - 4 \cdot 36 \cdot (-576)}}{2 \cdot 36} \]

10. **Calculating the Discriminant**:
Calculate \( b^2 - 4ac \):
\[ (-384)^2 = 147456 \]
\[ 4 \cdot 36 \cdot 576 = 82944 \]
\[ 147456 + 82944 = 230400 \]

11. **Finding the Square Root**:
\[ \sqrt{230400} = 480 \]

12. **Substituting Back**:
\[ d = \frac{384 \pm 480}{72} \]
This gives two potential solutions:
\[ d = \frac{864}{72} = 12 \quad \text{(valid)} \]
\[ d = \frac{-96}{72} = -\frac{4}{3} \quad \text{(not valid)} \]

Thus, the initial distance between the two bodies is:
\[ \boxed{12 \text{ meters}} \]