Problem 37 - Precalculus with Calculus Previews

Question

Two of the most devastating earthquakes in the San Francisco Bay area occurred in 1906 along the San Andreas fault and in 1989 in the Santa Cruz Mountains near Loma Prieta peak. The 1906 and 1989 earthquakes measured 8.5 and 7.1 on the Richter scale, respectively. How much greater was the intensity of the 1906 earthquake compared to the 1989 earthquake?

Step-by-Step Solution

Answer

The 1906 earthquake was approximately 25.12 times more intense than the 1989 earthquake.

Step 1: Understanding the Richter Scale

The Richter scale measures the magnitude of an earthquake, and it is a logarithmic scale. This means that each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy released.

Step 2: Formula for Intensity Ratio

To find how much greater the intensity of one earthquake is compared to another, we use the formula:\[ \text{Intensity Ratio} = 10^{(M_1 - M_2)} \]where \(M_1\) and \(M_2\) are the magnitudes of the two earthquakes.

Step 3: Substituting Values into the Formula

For the 1906 earthquake, \(M_1 = 8.5\), and for the 1989 earthquake \(M_2 = 7.1\). Substitute these values into the formula:\[ \text{Intensity Ratio} = 10^{(8.5 - 7.1)} \]

Step 4: Calculating the Intensity Ratio

Perform the calculation from the previous step:\[ \text{Intensity Ratio} = 10^{1.4} \approx 25.12 \]This means the 1906 earthquake was approximately 25.12 times more intense than the 1989 earthquake.

Key Concepts

logarithmic scale

In a logarithmic scale, each step represents a multiplication of a value. This is different from linear scales, where each step means an addition.
For example, on a linear scale, moving from 1 to 2 is simply an increase of 1. In contrast, on a logarithmic scale like the Richter Scale, moving from 1 to 2 indicates a multiplication by a consistent factor, which is 10 for the Richter Scale.
This type of scaling is useful in measuring phenomena with vast ranges in magnitude, such as sound intensity, light brightness, and earthquake magnitude. Here, each increase in number shows a tenfold increase in the measured amplitude.

• The nature of the scale helps in handling large variances efficiently.
• Logarithmic scales compress larger values, making them easier to compare.
• In the case of earthquakes, this compression makes intense quakes more manageable to plot and compare on a single scale.

intensity ratio

The intensity ratio helps describe how much more powerful one earthquake is compared to another. To find the intensity ratio on a logarithmic scale, the formula used is: \[ \text{Intensity Ratio} = 10^{(M_1 - M_2)} \] Here, \( M_1 \) and \( M_2 \) are the magnitudes of the two earthquakes you're comparing. This formula benefits largely from the properties of logarithms, which allow us to subtract two numbers in the exponent when comparing ratios.
Applying this insight to the 1906 and 1989 earthquakes, the formula shows precisely why the 1906 quake was more severe. The calculation:

• Substitute the magnitudes into the formula, \( 10^{(8.5 - 7.1)} \).
• The difference in magnitude, \( 8.5 - 7.1 \), is 1.4.
• Using the formula, we find the ratio to be \( 10^{1.4} \), approximately equaling 25.12.

Thus, the intensity of the 1906 earthquake was over 25 times greater than the 1989 earthquake.

earthquake magnitude

Earthquake magnitude describes the size or energy release of an earthquake. It's crucial to understand that this is not a direct measure of how devastating an earthquake will be, as that also depends on factors such as depth, location, and duration.
The magnitude itself is a number on the Richter scale, conveying the energy released at the earthquake's source. This concept came from the need to compare different earthquakes in terms of energy, using a standardized scale.

• Magnitude measures how much the ground shakes.
• Each whole number step on the Richter Scale accounts for 31.6 times more energy.
• This means a slight increase in magnitude represents a massive increase in potential damage.

The distinction between different earthquake magnitudes helps scientists and the public quickly understand the severity of seismic events. It is one part of a broader toolkit used for earthquake preparedness and response.