Problem 4 - Algebra 2

Question

Seismology In \(1812,\) an earthquake of magnitude 7.9 shook New Madrid, Missouri. Compare the amount of energy released by that earthquake to the amount of energy released by each earthquake below. magnitude 6.9 in Kobe, Japan, in 1995

Step-by-Step Solution

Answer

The New Madrid earthquake released approximately 31.6 times more energy than the Kobe earthquake.

Step 1: Understand the logarithmic relationship

Understand that the Richter scale is logarithmic. This means, for example, that an earthquake of magnitude 7.0 releases 10 times more energy than one of magnitude 6.0, and 100 times more energy than one of magnitude 5.0.

Step 2: Identify the given magnitudes

Identify the magnitudes of the two earthquakes. The New Madrid earthquake had a magnitude of 7.9 and the Kobe earthquake had a magnitude of 6.9.

Step 3: Calculate the ratio of energy released

Since the logarithmic relationship indicates that an increase in 1.0 magnitude is equivalent to approximately 31.6 times more energy released, the ratio of energy between the two earthquakes can be calculated simply as \(31.6^{(7.9-6.9)}\).

Step 4: Finding the answer

The energy released by the 7.9 earthquake is approximately 31.6 times greater than the energy released by the 6.9 earthquake.

Key Concepts

Logarithmic Relationship

The Richter scale is a tool used to measure the magnitude of earthquakes, and it operates on a logarithmic scale. This means each whole number increase on the scale represents a tenfold increase in measured amplitude and approximately 31.6 times more energy release.
This can be explained by logarithms, which are a way to express exponential relationships. A logarithm calculates how many times one number, the base, needs to be multiplied by itself to get another number. So, in seismic terms:

• An earthquake of magnitude 6 is 10 times more powerful, in terms of amplitude, than a 5.
• It also releases about 31.6 times more energy.

Understanding this logarithmic relationship helps us comprehend the immense differences in energy release between earthquakes of slightly different magnitudes.

Earthquake Magnitude

The magnitude of an earthquake quantifies the energy released at the source of the earthquake, and it is usually reported on the Richter scale. It's a crucial measure because it helps us understand the potential impact of seismic events.
Here are some critical points about earthquake magnitude:

• A higher magnitude indicates a more intense earthquake.
• The Richter scale, developed in 1935 by Charles F. Richter, was one of the first scales to use logarithms to compare the size of earthquakes.
• Each unit increase on the scale corresponds to much larger energy releases.

Scientists use seismographs to measure the amplitude of seismic waves, which then helps calculate the earthquake's magnitude based on the logarithmic scale. This process is crucial when comparing earthquakes of different magnitudes.

Energy Release Calculation

Calculating the energy released by an earthquake allows us to compare different seismic events. The energy release is correlated to the magnitude through a logarithmic formula.
Here's how you can understand this calculation:

• The energy release changes exponentially, approximately 31.6 times for each magnitude increase by 1.
• Given two earthquakes with different magnitudes, as in the New Madrid and Kobe example, we use the formula: \[ 31.6^{\Delta M} \]where \(\Delta M\) is the difference in magnitudes between the two earthquakes.
• In the case of New Madrid (7.9) and Kobe (6.9), the calculation is:\[ 31.6^{(7.9 - 6.9)} = 31.6^1 = 31.6 \]

The calculation shows that the New Madrid earthquake released about 31.6 times more energy than the Kobe earthquake.This approach helps us appreciate not just the scale of an earthquake, but its potential to cause damage and why small differences in magnitude are significant.