The Richter scale measures the magnitude of earthquakes based on the amplitude of the seismic waves. The formula for the Richter scale is given by:
\[ r = \log\left(\frac{A}{A_0}\right) \]
Where:
- \( r \) is the magnitude of the earthquake (in this case, 4.2),
- \( A \) is the amplitude of the seismic waves for the earthquake,
- \( A_0 \) is the amplitude of a reference seismic wave (the one that corresponds to a magnitude of 0).
We want to find out how many times stronger the amplitude \( A \) of the earthquake is compared to \( A_0 \). This means we need to find \( \frac{A}{A_0} \).
To find \( \frac{A}{A_0} \), we can rearrange the formula:
\[ r = \log\left(\frac{A}{A_0}\right) \implies \frac{A}{A_0} = 10^r \]
Substituting \( r = 4.2 \):
\[ \frac{A}{A_0} = 10^{4.2} \]
Now we can calculate \( 10^{4.2} \):
\[ 10^{4.2} \approx 15848.9319 \]
Thus,
\[ \frac{A}{A_0} \approx 15849 \]
This means that the wave amplitude of the earthquake was approximately 15,849 times stronger than the reference amplitude \( A_0 \).
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