Problem 48 - Calculus and its applications

Question

If two earthquakes have magnitudes \(R_{1}\) and \(R_{2},\) where \(\mathrm{R}_{1}>\mathrm{R}_{2},\) their relative intensity is given by \(R_{1}-R_{2}=\log \frac{I_{1}}{I_{2}}\). Thus, comparing an earthquake of magnitude 8.0 with another earthquake of magnitude \(5.5,\) we have \(8.0-5.5=\log \frac{I_{1}}{I_{2}},\) or \(2.5=\log \frac{I_{1}}{I_{2}},\) and \(I_{1}=10^{2.5} I_{2} .\) Since \(10^{2.5} \approx 316,\) an earthquake of magnitude 8.0 is about 316 times as intense as an earthquake of magnitude \(5.5 .\) Use this information. An earthquake of magnitude 5.0 is considered "moderate," with the potential to damage unreinforced structures. How many times more intense was the San Francisco earthquake of 1906 than a "moderately damaging" earthquake with a magnitude \(5.0 ?\)

Step-by-Step Solution

Answer

The San Francisco earthquake was about 794 times more intense than a moderate earthquake of magnitude 5.0.

Step 1: Understand the given problem

We are tasked with finding how many times more intense an earthquake of a higher magnitude (specifically, the San Francisco earthquake of 1906) is compared to a moderate earthquake of magnitude 5.0. The question follows a similar logic to the comparison provided in the example.

Step 2: Apply the formula for relative intensity

The given formula for relative intensity in earthquake magnitudes is \(R_{1}-R_{2} = \log \frac{I_{1}}{I_{2}}\). Here, \(R_1\) is the magnitude of the San Francisco earthquake, \(R_2\) is the magnitude of a moderate earthquake (5.0), and we need to find \(\frac{I_1}{I_2}\).

Step 3: Insert the magnitudes into the formula

The magnitude of the San Francisco earthquake in 1906 was approximately 7.9. Therefore, substitute \(R_1 = 7.9\) and \(R_2 = 5.0\) into the formula: \(7.9 - 5.0 = \log \frac{I_1}{I_2}\).

Step 4: Simplify the expression

Calculate the difference between the magnitudes: \(7.9 - 5.0 = 2.9\). Substitute back into the formula: \(2.9 = \log \frac{I_1}{I_2}\).

Step 5: Solve for intensity ratio

To find the intensity ratio \(\frac{I_1}{I_2}\), rewrite the expression in exponential form: \(\frac{I_1}{I_2} = 10^{2.9}\).

Step 6: Calculate the intensity ratio

Calculate \(10^{2.9}\) using a calculator: \(10^{2.9} \approx 794.33\). This means the San Francisco earthquake was approximately 794 times more intense than a moderate earthquake with a magnitude of 5.0.

Key Concepts

Logarithmic Scale

Understanding the logarithmic scale is crucial in the context of earthquake magnitudes because it gives us a way to measure and compare large variations in data with relative ease. Earthquakes are measured using the Richter scale, which is a logarithmic scale. This means that each increase of one point on the scale represents a tenfold increase in the measured amplitude of the earthquake's seismic waves. However, when it comes to energy release, an increase of one point equates to approximately 31.6 times more energy released.
This logarithmic nature helps us manage and understand the vast differences in earthquake sizes without unwieldy numbers. A small jump on the scale reflects a massive change in earthquake energy: a magnitude of 6.0 releases ten times the energy of a magnitude 5.0 earthquake.

• Logarithms transform multiplicative relationships into additive ones. This property is used in comparing magnitudes.
• Understanding this concept is key to grasping why small changes in magnitude can indicate large variations in earthquake intensity.

Magnitude Comparison

Magnitude comparison in earthquakes involves understanding the difference in magnitude values and interpreting these differences into a measure of intensity or energy expended. When we say an earthquake is 'stronger' due to a higher magnitude number, it refers to its relative intensity or energy released.
The formula used for this comparison is: \(R_1 - R_2 = \log \frac{I_1}{I_2}\). Here, \(R_1\) and \(R_2\) are the magnitudes of two different earthquakes, and \(I_1\) and \(I_2\) are their respective intensities. For example, if an earthquake has a magnitude of 8.0, and another of 5.5, the intensity comparison shows \(I_1 = 10^{2.5} I_2\), meaning the 8.0 magnitude earthquake is 316 times more intense than the 5.5 one.
This logarithmic relationship in magnitude comparison allows seismologists to convey the differences in earthquakes succinctly. By understanding the magnitude difference, one can understand the increase in destructive energy.

• The magnitude difference can be converted into an intensity ratio explaining how much more powerful an earthquake is in comparison to another.
• This concept is essential to assess the potential damage and scale of impact of earthquakes.

Seismic Intensity Calculation

Seismic intensity calculation involves using logarithms to determine how many times more intense one earthquake is compared to another using their magnitude values. In our example about the San Francisco earthquake of 1906 versus a moderate magnitude 5.0 earthquake, we employ the same formula: \(R_1 - R_2 = \log \frac{I_1}{I_2}\).
When calculating this for the San Francisco earthquake with a magnitude of 7.9, we compare it to a 5.0 magnitude event. Using the given formula: \(7.9 - 5.0 = \log \frac{I_1}{I_2}\), we end up with \(2.9 = \log \frac{I_1}{I_2}\). The next step is to express this in exponential form: \(\frac{I_1}{I_2} = 10^{2.9}\).
Calculating \(10^{2.9}\) gives us approximately 794.33, meaning the San Francisco earthquake was about 794 times more intense than a moderate earthquake of magnitude 5.0. This simple calculation not only highlights the power of seismic events but also the utility of logarithmic scales in interpreting natural phenomena.

• By using the logarithmic scale, intense natural events are represented in an understandable and measurable format.
• This method of calculation is pivotal for earthquake preparedness and historical earthquake comparison.