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?



Answer :

To compare the intensities of the two earthquakes based on their magnitudes, we follow the relationship given:

[tex]$ M = \log \left(\frac{I}{I_0}\right) $[/tex]

where [tex]\( M \)[/tex] is the magnitude, [tex]\( I \)[/tex] is the intensity, and [tex]\( I_0 \)[/tex] is the reference intensity.

First, recall that the logarithmic relationship can be rearranged to express intensity [tex]\( I \)[/tex] in terms of magnitude [tex]\( M \)[/tex]:

[tex]$ I = I_0 \cdot 10^M $[/tex]

Since [tex]\( I_0 \)[/tex] is the smallest detectable seismic activity and can be considered as 1 (for simplicity in our calculations), we have:

[tex]$ I = 10^M $[/tex]

### Step-by-Step Solution:

1. Calculate the intensity for the first earthquake:

- Magnitude of the first earthquake ([tex]\( M_1 \)[/tex]) = 8.6

Using the formula:

[tex]$ I_1 = 10^{8.6} $[/tex]

From the given information, we know:

[tex]$ I_1 \approx 398107170.5534969 $[/tex]

2. Calculate the intensity for the second earthquake:

- Magnitude of the second earthquake ([tex]\( M_2 \)[/tex]) = 8.2

Using the formula:

[tex]$ I_2 = 10^{8.2} $[/tex]

From the given information, we know:

[tex]$ I_2 \approx 158489319.2461111 $[/tex]

3. Calculate the factor by which the intensity of the first earthquake is greater than the intensity of the second earthquake:

We find this factor by dividing the intensity of the first earthquake by the intensity of the second earthquake:

[tex]$ \text{Factor} = \frac{I_1}{I_2} $[/tex]

Using the provided numerical values:

[tex]$ \text{Factor} \approx \frac{398107170.5534969}{158489319.2461111} \approx 2.511886431509582 $[/tex]

### Conclusion:

The intensity of the first earthquake (magnitude 8.6) was approximately 2.51 times greater than the intensity of the second earthquake (magnitude 8.2).