On April 11, 2012, two earthquakes were measured off the northwest coast of Sumatra. The first had a magnitude of 8.6. The second had a magnitude of 8.2. By what approximate factor was the intensity of the first earthquake greater than the intensity of the second earthquake?
[tex]M=\log \left(\frac{1}{t_0}\right)[/tex]
[tex]M=[/tex] the magnitude of an earthquake
I = the intensity of an earthquake
[tex]f_0=[/tex] the smallest seismic activity that can be measured, which is 1
A. 0.40
B. 1.02
C. 1.05
D. 2.51
Answer (1)
Use the formula M = lo g ( I ) to relate magnitude and intensity.
Find the intensities: I 1 = 1 0 8.6 and I 2 = 1 0 8.2 .
Calculate the ratio of intensities: I 2 I 1 = 1 0 8.6 − 8.2 = 1 0 0.4 .
The intensity of the first earthquake is approximately 1 0 0.4 ≈ 2.51 times greater than the second. 2.51
Explanation
Understanding the Problem We are given the magnitudes of two earthquakes, M 1 = 8.6 and M 2 = 8.2 . We want to find the factor by which the intensity of the first earthquake is greater than the intensity of the second earthquake. The formula relating magnitude and intensity is M = lo g ( I 0 I ) , where M is the magnitude, I is the intensity, and I 0 is the smallest seismic activity that can be measured, which is 1.
Applying the Formula Since I 0 = 1 , the formula simplifies to M = lo g ( I ) . Therefore, for the first earthquake, 8.6 = lo g ( I 1 ) , and for the second earthquake, 8.2 = lo g ( I 2 ) .
Finding the Intensities To find the intensities, we can rewrite the logarithmic equations in exponential form: I 1 = 1 0 8.6 and I 2 = 1 0 8.2 .
Calculating the Ratio We want to find the ratio of the intensities, which is I 2 I 1 = 1 0 8.2 1 0 8.6 . Using the properties of exponents, we have 1 0 8.2 1 0 8.6 = 1 0 8.6 − 8.2 = 1 0 0.4 .
Final Calculation and Answer Now we need to calculate 1 0 0.4 . The result of this calculation is approximately 2.51188643150958. Therefore, the intensity of the first earthquake is approximately 2.51 times greater than the intensity of the second earthquake.
Examples
Earthquakes release energy that can be measured using the Richter scale, which is logarithmic. This problem demonstrates how a small difference in magnitude can result in a significant difference in the intensity of the earthquake. In real life, this helps civil engineers design buildings that can withstand earthquakes of varying intensities, and it helps emergency responders prepare for the aftermath of such events. Understanding the scale and its implications can save lives and reduce damage.
