Problem 46 - Intermediate Algebra

Question

Calculate how many times more intense an earthquake with a Richter number of \(8.9\) is than an earthquake with a Richter number of \(6.2\).

Step-by-Step Solution

Answer

The earthquake with magnitude 8.9 is approximately 501 times more intense than the one with magnitude 6.2.

Step 1: Understanding the Richter Scale

The Richter scale is a logarithmic scale used to measure the intensity of an earthquake. Each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release.

Step 2: Formula for Intensity Comparison

To compare the intensity of two earthquakes on the Richter scale, we use the formula: \( I = 10^{M_1 - M_2} \) where \( M_1 \) and \( M_2 \) are the Richter magnitudes of the two earthquakes.

Step 3: Substitute Values into Formula

Substitute \( M_1 = 8.9 \) and \( M_2 = 6.2 \) into the formula: \[ I = 10^{8.9 - 6.2} \]

Step 4: Calculate the Exponent

First, calculate the difference between the two Richter magnitudes: \( 8.9 - 6.2 = 2.7 \).

Step 5: Compute the Intensity Multiple

Now, compute the intensity with the exponent 2.7: \[ I = 10^{2.7} \approx 501.2 \]. This means the earthquake with a magnitude of 8.9 is approximately 501 times more intense than one with a magnitude of 6.2.

Key Concepts

Logarithmic Scale

The Richter scale, used to measure earthquake magnitude, operates on a logarithmic scale. This means that each unit increase on the scale represents a tenfold increase in the wave amplitude. For instance, an earthquake measuring 5 on the Richter scale has waves that are ten times larger than those of an earthquake measuring 4. Additionally, the energy released by an earthquake increases about 31.6 times for each whole number increment.

Using a logarithmic scale allows us to convert the wide range of energies released by earthquakes into a manageable set of numbers. This makes comparisons clearer and more concise. By compressing the scale, we can more easily compare smaller earthquakes to larger, more destructive ones.

This mathematical approach helps to simplify complex data, enabling quick comprehension of an earthquake's potential impact. It provides a practical framework for scientists and emergency responders to assess situations quickly.

Earthquake Intensity

Earthquake intensity on the Richter scale doesn't just measure the shaking felt on the Earth's surface. Instead, it reflects the actual energy released by the earthquake, which is a more accurate indicator of potential damage.

Various factors contribute to the intensity, including:

• The depth of the earthquake's focus
• The geological conditions
• The distance from the epicenter

Understanding intensity is vital for assessing the possible effects on buildings and infrastructure. Areas closer to the epicenter often experience the most damage, but even distant locations can suffer if the intensity is high enough. This is why intensity is a critical aspect of earthquake analysis alongside magnitude.

Magnitude Comparison

Magnitude comparison refers to comparing the relative sizes of earthquakes to understand their differing impacts. With the Richter scale's logarithmic nature, a small difference in magnitude can represent a substantial difference in energy release.

For instance, when comparing two earthquakes with magnitudes of 8.9 and 6.2, we use the intensity formula:

• The formula is: \( I = 10^{M_1 - M_2} \)
• Substituting the given magnitudes: \( I = 10^{8.9 - 6.2} \)

The result is an intensity of about 501 times greater for the 8.9 magnitude earthquake. This demonstrates the significant scale difference caused by even small magnitude changes.

Magnitude comparison helps in prioritizing response and preparation efforts for different earthquakes, based on potential impact rather than just size.