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}{l_0}\right)[/tex]
[tex]M=[/tex] the magnitude of an earthquake
I = the intensity of an earthquake
[tex]I_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 express the intensities of the earthquakes in terms of their magnitudes: I 1 = 1 0 8.6 and I 2 = 1 0 8.2 .
Find the factor by which the intensity of the first earthquake is greater than the intensity of the second earthquake: F = I 2 I 1 = 1 0 8.2 1 0 8.6 = 1 0 0.4 .
Approximate the value of 1 0 0.4 , which is approximately 2.51.
The intensity of the first earthquake is approximately 2.51 times greater than the intensity of the second earthquake.
Explanation
Understanding the Problem We are given the magnitudes of two earthquakes, M 1 = 8.6 and M 2 = 8.2 , and we want to find the approximate factor by which the intensity of the first earthquake is greater than the intensity of the second. We are also given the formula M = lo g ( I 0 I ) , where M is the magnitude, I is the intensity, and I 0 = 1 is the smallest seismic activity that can be measured.
Calculating Intensities Since I 0 = 1 , the formula simplifies to M = lo g ( I ) . Therefore, I = 1 0 M . Let I 1 be the intensity of the first earthquake and I 2 be the intensity of the second earthquake. Then, I 1 = 1 0 M 1 = 1 0 8.6 and I 2 = 1 0 M 2 = 1 0 8.2 .
Finding the Factor We want to find the factor F = I 2 I 1 = 1 0 8.2 1 0 8.6 . Using the properties of exponents, we have F = 1 0 8.6 − 8.2 = 1 0 0.4 .
Approximating the Factor Now we need to approximate 1 0 0.4 . We can calculate this value. The result of the operation is approximately 2.51.
Final Answer 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 a logarithmic scale. This problem demonstrates how a small difference in magnitude can result in a significant difference in the intensity of the earthquake. Understanding the relative intensity of earthquakes helps engineers design safer buildings and helps emergency responders prepare for and respond to disasters more effectively. For example, an earthquake of magnitude 7 releases approximately 32 times more energy than an earthquake of magnitude 6, highlighting the importance of even small differences in magnitude.
