A and B started travelling towards each other at the same time, from places X to Y and Y to X, respectively. After crossing each other, A and B took 2.45 hours and 4.05 hours to reach Y and X, respectively. If the speed of B was 8.4 km/h, then what was the speed (in km/h) of A?

Options

A. 10.8
B. 9.9
C. 12.6
D. 11.7

Answer

To solve the problem, we will use the relationship between the speeds of A and B and the times they take to reach their respective destinations after crossing each other.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the speed of A be \( v_A \) km/h.
- The speed of B is given as \( v_B = 8.4 \) km/h.
- The time taken by A to reach Y after crossing B is \( t_A = 2.45 \) hours.
- The time taken by B to reach X after crossing A is \( t_B = 4.05 \) hours.

2. **Use the Speed-Time Relationship**:
- According to the problem, the ratio of the speeds of A and B is equal to the square root of the ratio of the times taken by them after crossing each other:
\[ \frac{v_A}{v_B} = \sqrt{\frac{t_B}{t_A}} \]

3. **Substitute the Known Values**:
- Substitute \( v_B = 8.4 \) km/h, \( t_A = 2.45 \) hours, and \( t_B = 4.05 \) hours into the equation:
\[ \frac{v_A}{8.4} = \sqrt{\frac{4.05}{2.45}} \]

4. **Calculate the Ratio of Times**:
- First, calculate \( \frac{4.05}{2.45} \):
\[ \frac{4.05}{2.45} = 1.65306122449 \quad (\text{approximately } 1.65) \]

5. **Calculate the Square Root**:
- Now, find the square root of \( 1.65 \):
\[ \sqrt{1.65} \approx 1.284 \]

6. **Set Up the Equation**:
- Now, substitute back into the speed ratio:
\[ \frac{v_A}{8.4} = 1.284 \]

7. **Solve for \( v_A \)**:
- Multiply both sides by \( 8.4 \) to find \( v_A \):
\[ v_A = 1.284 \times 8.4 \approx 10.78 \text{ km/h} \]

### Final Answer:
The speed of A is approximately **10.78 km/h**.