Determine the magnitude of an earthquake that is 200 times as intense as an earthquake of magnitude 3.7 on the Richter scale.
To determine the magnitude of an earthquake that is 200 times as intense as an earthquake of magnitude 3.7 on the Richter scale, we can use the logarithmic relationship between earthquake intensities.
The equation commonly used to relate earthquake intensities is:
I = A * 10^(B * M)
Where:
I = Intensity
A and B are constants
M = Magnitude
In this case, we can assume that the constant A = 1.0 (since it is not provided), and B = 1.5 (a commonly accepted constant for moderate earthquakes).
Now, let's substitute the given information into the equation and solve for M:
I = 200 * I(3.7)
200 = 1.0 * 10^(1.5 * M(3.7))
Dividing both sides by 1.0:
200/1.0 = 10^(1.5 * M(3.7))
Simplifying the equation:
200 = 10^(1.5 * 3.7M)
Taking the logarithm of both sides (base 10):
log(200) = (1.5 * 3.7M) * log(10)
Using a calculator, we can find that log(200) is approximately 2.301.
2.301 = 5.55M
Now, divide both sides by 5.55 to solve for M:
2.301 / 5.55 = M
M ≈ 0.41
Therefore, an earthquake that is 200 times as intense as an earthquake of magnitude 3.7 will have a magnitude of approximately 0.41 on the Richter scale.
To determine the magnitude of the earthquake that is 200 times as intense as an earthquake of magnitude 3.7 on the Richter scale, we can use the formula:
M2 = M1 + log10(A2/A1)
Where:
M2 = Magnitude of the earthquake we want to find
M1 = Magnitude of the given earthquake (3.7 in this case)
A2 = Amplitude of the earthquake we want to find
A1 = Amplitude of the given earthquake
In this case, we have the intensity ratio, which is 200 times. The intensity ratio is related to the amplitude ratio by the equation:
A2/A1 = 10^(1.5 * (M2 - M1))
To isolate M2, we rearrange the equation as follows:
M2 = M1 + log10(A2/A1)
In this case, M1 = 3.7 and A1 = 1 (since we are given that the magnitude of 3.7 corresponds to an intensity of 1 on the Richter scale).
Let's plug in the values into the equation and solve for M2:
M2 = 3.7 + log10(200)
Using a calculator, we find that:
M2 ≈ 3.7 + 2.3
M2 ≈ 5.99
Therefore, the magnitude of the earthquake that is 200 times as intense as an earthquake of magnitude 3.7 is approximately 5.99 on the Richter scale.