To determine the magnitude [tex]\( R \)[/tex] of an earthquake on the Richter scale, we use the formula:
[tex]\[ R = \log \left(\frac{A}{T}\right) + D \][/tex]
Here, let's break down the given parameters and the steps to find the magnitude:
1. Amplitude [tex]\( A \)[/tex]: The amplitude measured is [tex]\( A = 1 \)[/tex] micrometer.
2. Time [tex]\( T \)[/tex]: The duration of the earthquake is [tex]\( 4 \)[/tex] seconds.
3. Weakening Factor [tex]\( D \)[/tex]: The weakening factor, which depends on the distance from the epicenter, is [tex]\( 2 \)[/tex].
Using the provided parameters, we substitute these values into the formula:
[tex]\[ R = \log \left(\frac{1}{4}\right) + 2 \][/tex]
Next, we compute the logarithmic part of the equation:
- [tex]\(\frac{1}{4}\)[/tex] is the ratio of the amplitude [tex]\( A \)[/tex] to the time [tex]\( T \)[/tex].
- We then take the base-10 logarithm (logarithm to the base 10) of this ratio.
[tex]\[ \log \left(\frac{1}{4}\right) \][/tex]
Calculating [tex]\(\log \frac{1}{4}\)[/tex]:
1. [tex]\(\frac{1}{4} = 0.25\)[/tex]
2. [tex]\(\log(0.25) \approx -0.60206\)[/tex]
Now we add the weakening factor [tex]\( D = 2 \)[/tex] to this log value:
[tex]\[ R = -0.60206 + 2 \][/tex]
Perform the addition:
[tex]\[ R = 1.39794 \][/tex]
So, the magnitude [tex]\( R \)[/tex] of the earthquake is approximately [tex]\( 1.39794 \)[/tex].
In conclusion, when plotted on a graph of the Richter scale, the earthquake with the given parameters would show a magnitude of approximately [tex]\( 1.39794 \)[/tex].
