Problem 10 - A Graphical Approach to College Algebra

Question

The earthquake off the coast of Northern Sumatra on Dec. \(26,2004,\) had a Richter scale rating of 8.9 (a) Express the intensity of this earthquake in terms of \(I_{0}\). (b) Aftershocks from this quake had a Richter scale rating of \(6.0 .\) Express the intensity of these in terms of \(I_{0}\) (c) Compare the intensities of the 8.9 earthquake to the 6.0 aftershock.

Step-by-Step Solution

Answer

(a) Intensity is \( I_0 \times 10^{8.9} \). (b) Intensity is \( I_0 \times 10^6 \). (c) Main earthquake is \( 10^{2.9} \) times more intense.

Step 1: Understanding the Richter Scale

The Richter scale is logarithmic. The magnitude of an earthquake is expressed as \( M = \log_{10} \left( \frac{I}{I_0} \right) \) where \( I \) is the intensity of the earthquake and \( I_0 \) is a reference intensity.

Step 2: Finding the Intensity for Main Earthquake

Given the magnitude of the earthquake (8.9), we use the formula \( 8.9 = \log_{10} \left( \frac{I}{I_0} \right) \). Solving for \( I \), we have \( I = I_0 \times 10^{8.9} \). So, the intensity of the main earthquake is \( I = I_0 \times 10^{8.9} \).

Step 3: Finding the Intensity for Aftershock

The magnitude of the aftershock is given as 6. Using the formula \( 6.0 = \log_{10} \left( \frac{I}{I_0} \right) \), we solve for \( I \) to find \( I = I_0 \times 10^{6} \). Thus, the intensity of the aftershock is \( I = I_0 \times 10^6 \).

Step 4: Comparing the Intensities

To compare the intensities, divide the intensity of the main earthquake by the intensity of the aftershock: \( \frac{I_{main}}{I_{aftershock}} = \frac{I_0 \times 10^{8.9}}{I_0 \times 10^6} = 10^{8.9 - 6} = 10^{2.9} \). Therefore, the main earthquake is \( 10^{2.9} \) times more intense than the aftershock.

Key Concepts

Logarithmic Scale

The Richter scale is an example of a logarithmic scale, which is commonly used for measuring extremely large or small quantities. A logarithmic scale is different from a linear scale because each step on a logarithmic scale is a multiplication of the previous step. In contrast, a linear scale increases by addition. For example, on the Richter scale, a magnitude of 7.0 means an earthquake is ten times more intense than one measured at 6.0. This scaling is expressed mathematically by the formula: \[M = \log_{10} \left( \frac{I}{I_0} \right)\]Where:

• \( M \) is the magnitude on the Richter scale,
• \( I \) is the intensity of the earthquake,
• \( I_0 \) is the reference intensity used for comparison.

Thus, when earthquake magnitudes are expressed in terms of logarithmic scale, small increases in magnitude represent substantial increases in actual intensity.

Earthquake Intensity Comparison

Comparing earthquake intensities involves examining the difference in their magnitudes on the logarithmic Richter scale. The intensity of an earthquake is linked to the amount of energy it releases. To compare two earthquakes, you subtract their magnitudes. The result tells you how many times more intense one is compared to the other. For instance, given:

• A main earthquake with a magnitude of 8.9
• An aftershock with a magnitude of 6.0

The difference is\[8.9 - 6.0 = 2.9\]This indicates that the main earthquake is \(10^{2.9}\) times more intense. Understanding this comparison helps assess the impact and potential damage caused by different seismic events. Even a small difference in magnitude can mean a vastly more destructive earthquake.

Richter Scale Calculations

Calculating the intensity of an earthquake using the Richter scale involves a straightforward process of applying logarithmic functions. Given the magnitudes:- For the main earthquake rated 8.9, the intensity calculation involves: \[ I = I_0 \times 10^{8.9} \] This is how much stronger or weaker an earthquake is compared to a base level.- For an aftershock rated 6.0, the calculation is: \[ I = I_0 \times 10^{6} \]The comparison of both intensities can be achieved by dividing one by the other:\[\frac{I_{main}}{I_{aftershock}} = \frac{I_0 \times 10^{8.9}}{I_0 \times 10^{6}} = 10^{2.9}\]This tells us that the main earthquake is significantly more intense than the aftershock. These calculations are crucial for understanding the relative power of different seismic events, helping scientists and safety officials to better prepare and respond to earthquakes.