Problem 47 - 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. The following table shows the magnitudes of selected large earthquakes. $$ \begin{array}{|l|c|} \hline \text { EARTHQUAKE } & \text { MAGNITUDE } \\ \hline \text { Sumatran-Andaman, } 2004 & 9.2 \\ \text { Japan, } 2011 & 9.0 \\ \text { San Francisco, } 1906 & 8.0 \\ \text { Baja California, } 2010 & 7.2 \\ \text { San Fernando, } 1971 & 6.6 \\ \hline \end{array} $$ a) How many times more intense was the Japanese earthquake of 2011 than the Baja California earthquake of \(2010 ?\) b) How many times more intense was the SumatranAndaman earthquake of 2004 than the San Fernando earthquake of \(1971 ?\)

Step-by-Step Solution

Answer

a) About 63 times; b) About 398 times.

Step 1: Understand the Formula

The formula provided is \( R_1 - R_2 = \log \frac{I_1}{I_2} \), where \( R_1 \) and \( R_2 \) are magnitudes of two earthquakes, and \( I_1 \) and \( I_2 \) are their respective intensities. The goal is to find \( \frac{I_1}{I_2} \).

Step 2: Calculate the Magnitude Difference for Part (a)

For the Japanese earthquake (2011), \( R_1 = 9.0 \), and for the Baja California earthquake (2010), \( R_2 = 7.2 \). The difference is \( 9.0 - 7.2 = 1.8 \).

Step 3: Solve for Relative Intensity for Part (a)

Substitute the difference into the formula: \( 1.8 = \log \frac{I_1}{I_2} \). To find \( \frac{I_1}{I_2} \), calculate \( 10^{1.8} \), which is approximately 63.1.

Step 4: Calculate the Magnitude Difference for Part (b)

For the Sumatran-Andaman earthquake (2004), \( R_1 = 9.2 \), and for the San Fernando earthquake (1971), \( R_2 = 6.6 \). The difference is \( 9.2 - 6.6 = 2.6 \).

Step 5: Solve for Relative Intensity for Part (b)

Substitute the difference into the formula: \( 2.6 = \log \frac{I_1}{I_2} \). Calculate \( 10^{2.6} \), which is approximately 398.1.

Key Concepts

Logarithmic Formulas

Understanding logarithmic formulas is crucial when working with earthquake magnitudes. A logarithmic scale is a nonlinear scale used to compare quantities that differ greatly in magnitude. In the context of earthquakes, this is incredibly useful because it allows us to compare the intensity of two earthquakes using a manageable scale. The key formula used here is:\[R_1 - R_2 = \log \frac{I_1}{I_2}\]In this formula, \(R_1\) and \(R_2\) represent the magnitudes of two different earthquakes. The difference in magnitude is equal to the logarithm of the ratio of their intensities, \(\frac{I_1}{I_2}\). This means that the intensity difference can be expressed in terms of a power of 10. A small difference in magnitude can represent a significant difference in actual intensity. This highlights the strength of logarithmic scales in handling large numerical differences.

Intensity Comparison

When comparing the intensity of two earthquakes, the formula \(R_1 - R_2 = \log \frac{I_1}{I_2}\) can be directly applied to find how many times more intense one earthquake is compared to another. For example, if one earthquake has a magnitude \(R_1\) of 9.0 and another earthquake has a magnitude \(R_2\) of 7.2, the magnitude difference is 1.8. Using the formula: \[ 9.0 - 7.2 = \log \frac{I_1}{I_2} \]the solution becomes solving for \( \frac{I_1}{I_2} \), which is equal to \(10^{1.8}\). Calculating \(10^{1.8}\) yields approximately 63.1, meaning that the earthquake of magnitude 9.0 is about 63.1 times as intense as the earthquake of magnitude 7.2.
Such insights allow scientists and researchers to quickly assess the potential damage and alert systems appropriately.

Magnitude Differences

Magnitude differences in earthquakes are more than just numerical anomalies—they reflect the exponentially scaling nature of earthquake intensity. For a difference of 2.5 in magnitudes, the intensity comparison derives from solving: \[2.5 = \log \frac{I_1}{I_2} \]This results in a calculation of \(10^{2.5}\) which translates to approximately 316 times greater intensity. In our example exercise, when dealing with the Sumatran-Andaman earthquake with magnitude 9.2 and the San Fernando earthquake with magnitude 6.6, the magnitude difference was 2.6. Plugging this into our formula:\[ 2.6 = \log \frac{I_1}{I_2} \]Solving gives \(10^{2.6} \approx 398.1\). This shows the earthquake of magnitude 9.2 was roughly 398 times more intense than the earthquake with a magnitude of 6.6.
These computations illustrate how even small differences in measured magnitudes correspond to vast differences in experienced intensity.