Final answer:
To calculate the distance to the earthquake's epicenter, we use the known speeds of P waves (8.7 km/s) and S waves (4.2 km/s) and their arrival time difference (1.9 minutes or 114 seconds). By setting up an equation and solving for the distance, we find that the epicenter is approximately 924.708 kilometers away from the observation station.
Step-by-step explanation:
To determine how far away the earthquake's epicenter is from the observation station, we can use the given speeds of the P waves and the S waves, and the time interval between their detection by the station. Since the P waves travel at 8.7 km/s and the S waves at 4.2 km/s, and the interval of their arrival is 1.9 minutes (which is 114 seconds), we can set up the following relationship:
Let d be the distance to the epicenter. The P waves take d/8.7 seconds to reach the station, and the S waves take d/4.2 seconds. The time difference between their arrivals is given by:
(d/4.2) - (d/8.7) = 114 seconds
Solving for d:
d[(1/4.2) - (1/8.7)] = 114
d(8.7 - 4.2) / (4.2 × 8.7) = 114
d(4.5) / (36.54) = 114
d = 114 × 36.54 / 4.5
d = 924.708 km
The earthquake's epicenter is approximately 924.708 kilometers away from the observation station.