To determine the factor by which the intensity of an earthquake with magnitude 5.4 is greater than an earthquake with magnitude 5.3, we need to first understand how the magnitude and intensity of an earthquake are related.
The Richter scale defines the magnitude of an earthquake (M) as a logarithmic function of its intensity (I), where:
[tex]\[ M = \log \left(\frac{I}{I_0}\right) \][/tex]
Here, [tex]\( I_0 \)[/tex] is the smallest measurable seismic activity, which is given as 1. We can simplify the formula to:
[tex]\[ I = 10^M \][/tex]
This means the intensity (I) grows exponentially with each unit increase in magnitude (M).
Let's calculate the intensities for both earthquakes:
1. Intensity of the earthquake with magnitude 5.4:
[tex]\[
I_1 = 10^{5.4}
\][/tex]
From the provided solution, we know:
[tex]\[
I_1 \approx 251188.643
\][/tex]
2. Intensity of the earthquake with magnitude 5.3:
[tex]\[
I_2 = 10^{5.3}
\][/tex]
From the provided solution, we know:
[tex]\[
I_2 \approx 199526.231
\][/tex]
Next, we find the factor by which the intensity of the earthquake with magnitude 5.4 is greater than the intensity of the earthquake with magnitude 5.3:
[tex]\[
\text{Intensity Factor} = \frac{I_1}{I_2}
\][/tex]
Plugging in the values:
[tex]\[
\text{Intensity Factor} = \frac{251188.643}{199526.231} \approx 1.2589254118
\][/tex]
Given the choices:
- 1.01
- 1.21
- 1.26
- 10.44
The closest approximate value to 1.2589254118 is:
[tex]\[
1.26
\][/tex]
Thus, the intensity of an earthquake with magnitude 5.4 is approximately 1.26 times greater than that of an earthquake with magnitude 5.3.
