Question
The Richter scale measures the magnitude, M, of an earthquake as a function of its intensity, I, and the intensity of a reference earthquake,
.
Which equation calculates the magnitude of an earthquake with an intensity 10,000 times that of the reference earthquake?
.
Which equation calculates the magnitude of an earthquake with an intensity 10,000 times that of the reference earthquake?
Answers
GPT-4o mini
The Richter scale is defined by the formula:
\[
M = \log_{10}\left(\frac{I}{I_0}\right)
\]
where \( M \) is the magnitude of the earthquake, \( I \) is the intensity of the earthquake being measured, and \( I_0 \) is the intensity of a reference earthquake.
If the intensity of the earthquake is 10,000 times that of the reference earthquake, we can express this relationship as:
\[
I = 10,000 \times I_0
\]
Now, substituting this into the Richter scale formula gives:
\[
M = \log_{10}\left(\frac{10,000 \times I_0}{I_0}\right)
\]
The \( I_0 \) cancels out:
\[
M = \log_{10}(10,000)
\]
Since \( 10,000 \) is equal to \( 10^4 \):
\[
M = \log_{10}(10^4) = 4
\]
Therefore, the magnitude \( M \) of an earthquake with an intensity 10,000 times that of the reference earthquake is \( 4 \).
\[
M = \log_{10}\left(\frac{I}{I_0}\right)
\]
where \( M \) is the magnitude of the earthquake, \( I \) is the intensity of the earthquake being measured, and \( I_0 \) is the intensity of a reference earthquake.
If the intensity of the earthquake is 10,000 times that of the reference earthquake, we can express this relationship as:
\[
I = 10,000 \times I_0
\]
Now, substituting this into the Richter scale formula gives:
\[
M = \log_{10}\left(\frac{10,000 \times I_0}{I_0}\right)
\]
The \( I_0 \) cancels out:
\[
M = \log_{10}(10,000)
\]
Since \( 10,000 \) is equal to \( 10^4 \):
\[
M = \log_{10}(10^4) = 4
\]
Therefore, the magnitude \( M \) of an earthquake with an intensity 10,000 times that of the reference earthquake is \( 4 \).