Final answer:
The distance to the epicenter of the earthquake can be calculated using the known velocities of P-waves and S-waves and their arrival time difference. Solving the set equations for time and substituting back to find distance, the earthquake was approximately 872.12 kilometers away from the seismograph.
Step-by-step explanation:
The question is asking how to find the distance to the epicenter of an earthquake given the velocity of transverse waves (S-waves), the velocity of longitudinal waves (P-waves), and the time difference in their arrival at a seismograph. To solve this, we can use the relationship d = vt, where d is the distance traveled, v is the velocity, and t is the time. The distance to the earthquake can be found by setting up two equations that account for the different velocities of P-waves and S-waves and their time difference upon arrival.
Let's denote the time it took for the P-waves to reach the seismograph as t. Then, the time it took for the S-waves will be t + 73s (since they arrived 73 seconds later). Assuming the distance to the earthquake is the same for both types of waves and using the provided velocities:
- Distance for P-waves is d = 5.1 km/s × t
- Distance for S-waves is d = 8.9 km/s × (t + 73s)
By setting these equal to each other, we solve for t and then substitute it back to get the distance d. This yields:
5.1t = 8.9(t + 73)
5.1t = 8.9t + 650.7
3.8t = 650.7
t = 650.7 / 3.8
t ≈ 171.2 seconds
Now, using t to find d:
d = 5.1 km/s × 171.2s
d ≈ 872.12 km
Therefore, the earthquake was approximately 872.12 kilometers away from the seismograph station.