A seismograph 300 km from the epicenter of an earthquake recorded a maximum amplitude of 5.7 102 µm. Find this earthquake's magnitude on the Richter scale. (Round your answer to the nearest tenth.)
To find the earthquake's magnitude on the Richter scale, we need to use the formula:
M = log10(A / A0)
Where:
M = magnitude on the Richter scale
A = maximum amplitude recorded by the seismograph (in µm)
A0 = reference amplitude (usually 1 µm)
In this case, we are given:
A = 5.7 * 10^2 µm = 570 µm
Now, we need to determine the reference amplitude, A0. The reference amplitude is the amplitude of a magnitude 0 earthquake measured 100 km from the epicenter.
In this problem, the seismograph is 300 km from the epicenter, so we need to adjust the reference amplitude using a distance correction.
The distance correction factor, D, is given by the formula:
D = 10^((3/2) * (log10(R) - 2))
Where:
R = distance from the epicenter to the seismograph (in km)
In this case, R = 300 km, so we can calculate D:
D = 10^((3/2) * (log10(300) - 2))
≈ 1.26
Now, we can calculate the adjusted reference amplitude, A0:
A0 = 1 µm * D
≈ 1 µm * 1.26
≈ 1.26 µm
Finally, we can substitute the values of A and A0 into the formula to find the magnitude, M:
M = log10(A / A0)
= log10(570 µm / 1.26 µm)
≈ log10(452.38)
≈ 2.655 (rounded to the nearest tenth)
Therefore, the magnitude of the earthquake on the Richter scale is approximately 2.7 (rounded to the nearest tenth).