An earthquake generates both transverse (S) and longitudinal (P) sound waves in the earth. The speed of S waves is about 4.5 km /s and that of P waves is about 8.0 km/s . A seismograph records P and S waves from an earthquake. The first P wave arrives 4.0 min before the first S wave. The epicenter of the earthquake is located at a distance about

Options

A. 25km
B. 250km
C. 2500km
D. 5000km

Explanation

To solve the problem, we need to find the distance to the epicenter of the earthquake based on the arrival times of the P and S waves.

### Step-by-Step Solution:

1. **Understanding Wave Speeds**:
- The speed of P waves (longitudinal) is given as \( V_P = 8.0 \, \text{km/s} \).
- The speed of S waves (transverse) is given as \( V_S = 4.5 \, \text{km/s} \).

2. **Setting Up the Relationship**:
- Let \( T_S \) be the time taken by the S wave to reach the seismograph.
- Let \( T_P \) be the time taken by the P wave to reach the seismograph.
- According to the problem, the P wave arrives 4 minutes before the S wave:
\[ T_S - T_P = 4 \, \text{minutes} = 240 \, \text{seconds} \]

3. **Distance Calculation**:
- The distance to the epicenter (D) can be expressed in terms of the time and speed of the waves:
\[ D = V_P \cdot T_P = V_S \cdot T_S \]
- Therefore, we can write:
\[ D = 8.0 \, T_P \quad \text{(1)} \]
\[ D = 4.5 \, T_S \quad \text{(2)} \]

4. **Expressing \( T_P \) in terms of \( T_S \)**:
- From the time difference equation, we can express \( T_P \) as:
\[ T_P = T_S - 240 \]

5. **Substituting \( T_P \) into Equation (1)**:
- Substitute \( T_P \) into equation (1):
\[ D = 8.0 \, (T_S - 240) \]
- Expanding this gives:
\[ D = 8.0 \, T_S - 1920 \quad \text{(3)} \]

6. **Setting Equations (2) and (3) Equal**:
- Since both equations represent the same distance \( D \):
\[ 4.5 \, T_S = 8.0 \, T_S - 1920 \]

7. **Solving for \( T_S \)**:
- Rearranging gives:
\[ 1920 = 8.0 \, T_S - 4.5 \, T_S \]
\[ 1920 = 3.5 \, T_S \]
\[ T_S = \frac{1920}{3.5} \approx 548.57 \, \text{seconds} \]

8. **Calculating Distance \( D \)**:
- Now, substitute \( T_S \) back into equation (2):
\[ D = 4.5 \, T_S = 4.5 \times 548.57 \approx 2464.57 \, \text{km} \]
- Rounding this gives approximately \( D \approx 2500 \, \text{km} \).

### Final Answer:
The epicenter of the earthquake is located at a distance of approximately **2500 km**.