Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse (S) and longitudinal (P) sound waves. Typically, the speed of S wave is about 4.0 km `s^(-1)`, and that of P wave is 8.0 `kms^(-1)`. A seismograph records P and S waved from an earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, how far away does the earthquake occur?

Options

A. 192 km
B. 384 km
C. 1920 km
D. 3840 km

Explanation

To solve the problem of determining how far away the earthquake occurred based on the arrival times of P and S waves, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data**:
- Speed of S wave, \( V_S = 4 \, \text{km/s} \)
- Speed of P wave, \( V_P = 8 \, \text{km/s} \)
- Time difference between arrival of P wave and S wave, \( \Delta T = 4 \, \text{minutes} = 240 \, \text{seconds} \)

2. **Define Variables**:
- Let \( T_P \) be the time taken by the P wave to reach the seismograph.
- Let \( T_S \) be the time taken by the S wave to reach the seismograph.

3. **Establish Relationships**:
- From the problem, we know that:
\[ T_S = T_P + 240 \, \text{seconds} \]

4. **Use the Speed-Relation Formula**:
- The distance \( L \) traveled by both waves can be expressed as:
\[ L = V_P \cdot T_P \quad \text{(for P wave)} \]
\[ L = V_S \cdot T_S \quad \text{(for S wave)} \]

5. **Set the Two Equations Equal**:
- Since both expressions equal \( L \), we can set them equal to each other:
\[ V_P \cdot T_P = V_S \cdot T_S \]

6. **Substitute for \( T_S \)**:
- Substitute \( T_S \) from step 3 into the equation:
\[ 8 \cdot T_P = 4 \cdot (T_P + 240) \]

7. **Solve for \( T_P \)**:
- Distributing the right side:
\[ 8T_P = 4T_P + 960 \]
- Rearranging gives:
\[ 8T_P - 4T_P = 960 \]
\[ 4T_P = 960 \]
\[ T_P = 240 \, \text{seconds} \]

8. **Calculate \( T_S \)**:
- Now, substitute \( T_P \) back to find \( T_S \):
\[ T_S = T_P + 240 = 240 + 240 = 480 \, \text{seconds} \]

9. **Calculate Distance \( L \)**:
- Use either wave's formula to find \( L \). Using \( V_P \):
\[ L = V_P \cdot T_P = 8 \cdot 240 = 1920 \, \text{km} \]

### Final Answer:
The distance from the earthquake to the seismograph is **1920 km**.

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