A seismograph 300 km from the epicenter of an earthquake recorded a maximum amplitude of 5.5 multiplied by 102 µm. Find this earthquake's magnitude on the Richter scale. (Round your answer to the nearest tenth.)
M =
To find the earthquake's magnitude on the Richter scale, we need to use the formula:
M = log(A/T)
Where:
M = Magnitude on the Richter scale
A = Maximum amplitude of the seismic waves recorded by the seismograph (in µm)
T = Period in seconds (the time it takes for the waves to complete one cycle)
In this case, we are given the maximum amplitude (A) as 5.5 * 10^2 µm.
To find the period (T), we can use the following formula:
T = 10^(0.123 + 1.44M)
Where M is the magnitude. We can rearrange this formula to solve for M:
M = (log10(T) - 0.123) / 1.44
Let's substitute the given values and calculate the magnitude:
T = 10^(0.123 + 1.44M)
T = 10^(0.123 + 1.44 * M)
T = 10^0.123 * 10^(1.44 * M)
T ≈ 1.127 * 10^(1.44 * M)
Since we know that T is the time it takes for the waves to complete one cycle, it is a measure of the duration of the earthquake. It is unlikely to be provided in this question, so we can use an average value for T.
A typical value for T is 1 second. So, let's use T = 1 second.
Now we can substitute T = 1 in the equation:
1 = 1.127 * 10^(1.44 * M)
To solve for M, we need to isolate it on one side of the equation.
Divide both sides of the equation by 1.127:
1/1.127 = 10^(1.44 * M)
0.887 = 10^(1.44 * M)
Now take the logarithm of both sides (base 10):
log10(0.887) = log10(10^(1.44 * M))
log10(0.887) = 1.44 * M
Now solve for M by dividing both sides by 1.44:
M = log10(0.887) / 1.44
Using a calculator or a math software, we can find:
M ≈ -0.0592
Rounding this value to the nearest tenth:
M ≈ -0.1
Therefore, the earthquake's magnitude on the Richter scale is approximately -0.1.