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

Answer

To find the distance of the epicenter of the earthquake from the seismograph, we can follow these steps:

### Step-by-Step Solution

1. **Define Variables:**
- Let \( V_P \) be the speed of P waves (longitudinal) = 8.0 km/s.
- Let \( V_S \) be the speed of S waves (transverse) = 4.5 km/s.
- Let \( D \) be the distance to the epicenter.
- Let \( T_P \) be the time taken for the P wave to reach the seismograph.
- Let \( T_S \) be the time taken for the S wave to reach the seismograph.

2. **Relate Distance, Speed, and Time:**
- The distance \( D \) can be expressed in terms of the speeds and times:
\[ D = V_P \cdot T_P = V_S \cdot T_S \]

3. **Express \( T_P \) in terms of \( T_S \):**
- Since the P wave arrives 4 minutes (or 240 seconds) before the S wave, we can write:
\[ T_S - T_P = 240 \text{ seconds} \]
- Rearranging gives:
\[ T_P = T_S - 240 \]

4. **Substitute \( T_P \) into the distance equation:**
- From the distance equation, we have:
\[ D = V_P \cdot (T_S - 240) \]
- Substituting the values of \( V_P \) and \( D \):
\[ D = 8.0 \cdot (T_S - 240) \]

5. **Set the two distance equations equal:**
- We also have:
\[ D = V_S \cdot T_S = 4.5 \cdot T_S \]
- Setting the two expressions for \( D \) equal to each other:
\[ 8.0 \cdot (T_S - 240) = 4.5 \cdot T_S \]

6. **Solve for \( T_S \):**
- Expanding the left side:
\[ 8.0 T_S - 1920 = 4.5 T_S \]
- Rearranging gives:
\[ 8.0 T_S - 4.5 T_S = 1920 \]
\[ 3.5 T_S = 1920 \]
\[ T_S = \frac{1920}{3.5} \approx 548.57 \text{ seconds} \]

7. **Calculate the distance \( D \):**
- Now substitute \( T_S \) back into the equation for \( D \):
\[ D = 4.5 \cdot T_S = 4.5 \cdot 548.57 \approx 2468.57 \text{ km} \]
- Rounding gives:
\[ D \approx 2500 \text{ km} \]

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