What approximate factor is the intensity of an earthquake with magnitude 5.4 greater than an earthquake with magnitude 5.3?
[tex]M=\log \left(\frac{1}{l_0}\right)[/tex]
[tex]M=[/tex] the magnitude of an earthquake
[tex]I=[/tex] the intensity of an earthquake
[tex]I_0=[/tex] the smallest seismic activity that can be measured, which is 1
Answer (1)
The problem uses the formula M = lo g ( I 0 I ) to relate earthquake magnitude and intensity.
We determine the ratio of intensities I 2 I 1 for magnitudes 5.4 and 5.3.
Calculate the ratio as 1 0 5.4 − 5.3 = 1 0 0.1 .
Approximate the value of 1 0 0.1 to find the factor: 1.26 .
Explanation
Understanding the Problem We are given the formula M = lo g ( I 0 I ) , where M is the magnitude of an earthquake, I is the intensity of the earthquake, and I 0 is the smallest seismic activity that can be measured, which is 1. We want to find the approximate factor by which the intensity of an earthquake with magnitude 5.4 is greater than an earthquake with magnitude 5.3.
Setting up the Equations Let M 1 = 5.4 and M 2 = 5.3 be the magnitudes of the two earthquakes. Let I 1 and I 2 be the intensities of the two earthquakes, respectively. We have M 1 = lo g ( I 1 / I 0 ) and M 2 = lo g ( I 2 / I 0 ) . Since I 0 = 1 , we have M 1 = lo g ( I 1 ) and M 2 = lo g ( I 2 ) .
Finding the Ratio of Intensities We want to find the factor I 2 I 1 . We have I 1 = 1 0 M 1 and I 2 = 1 0 M 2 . Therefore, I 2 I 1 = 1 0 M 2 1 0 M 1 = 1 0 M 1 − M 2 .
Substituting the Values Substituting M 1 = 5.4 and M 2 = 5.3 , we get I 2 I 1 = 1 0 5.4 − 5.3 = 1 0 0.1 .
Calculating the Factor Calculating 1 0 0.1 , we find that 1 0 0.1 ≈ 1.2589 . Rounding to two decimal places, we get approximately 1.26.
Final Answer Therefore, the intensity of an earthquake with magnitude 5.4 is approximately 1.26 times greater than the intensity of an earthquake with magnitude 5.3.
Examples
Earthquakes release energy that can be measured using the Richter scale, which is a logarithmic scale. This problem demonstrates how a small difference in magnitude can result in a significant difference in the intensity of the earthquake. For example, understanding this difference is crucial for engineers designing buildings in earthquake-prone areas, as even a seemingly small increase in expected earthquake magnitude can necessitate significantly stronger building designs. This knowledge also helps in disaster preparedness and risk assessment.
