An earthquake generates both transverse (S) and longitudinal (P) sound waves in the earth. The speed of S waves is about 6 km `s^(-1)` and that of P waves is about 9 km `s^(-1)`. A seismograph records P and S waves from an earthquarke. The first P wave arrives 5 minutes before the first S wave. the epicenter of the earth quake is located at a distance

Options

A. 54 km
B. 540 km
C. 5400 km
D. 72 km

Answer

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. **Define Variables:**
Let the distance to the epicenter be \( x \) kilometers.

2. **Calculate Time Taken by Each Wave:**
- The speed of S waves (transverse waves) is \( 6 \, \text{km/s} \).
- The speed of P waves (longitudinal waves) is \( 9 \, \text{km/s} \).
- The time taken by the S wave to travel distance \( x \) is given by:
\[ T_S = \frac{x}{6} \]
- The time taken by the P wave to travel distance \( x \) is given by:
\[ T_P = \frac{x}{9} \]

3. **Set Up the Equation Based on Arrival Times:**
According to the problem, the first P wave arrives 5 minutes before the first S wave. We can express this as:
\[ T_S - T_P = 5 \, \text{minutes} \]
Converting 5 minutes to seconds:
\[ 5 \, \text{minutes} = 5 \times 60 = 300 \, \text{seconds} \]
Therefore, we have:
\[ \frac{x}{6} - \frac{x}{9} = 300 \]

4. **Solve the Equation:**
To solve the equation, we need a common denominator. The least common multiple of 6 and 9 is 18. Rewriting the equation:
\[ \frac{3x}{18} - \frac{2x}{18} = 300 \]
Simplifying the left side:
\[ \frac{3x - 2x}{18} = 300 \]
This simplifies to:
\[ \frac{x}{18} = 300 \]
Multiplying both sides by 18:
\[ x = 300 \times 18 \]
Calculating the right side:
\[ x = 5400 \, \text{km} \]

5. **Conclusion:**
The distance to the epicenter of the earthquake is \( 5400 \, \text{km} \).

### Final Answer:
The epicenter of the earthquake is located at a distance of **5400 kilometers**.

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