Problem 39 - Precalculus with Calculus Previews

Question

If an earthquake has a magnitude 4.2 on the Richter scale, what is the magnitude on the Richter scale of an earthquake that has an intensity 20 times greater?

Step-by-Step Solution

Answer

The second earthquake has a magnitude of 5.5 on the Richter scale.

Step 1: Understand the Richter Scale

The magnitude on the Richter scale is logarithmic, meaning each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release. The formula for the Richter scale is \( M = \log_{10}(I/I_0) \), where \( M \) is the magnitude, \( I \) is the intensity of the earthquake, and \( I_0 \) is a reference intensity.

Step 2: Set Up the Equation for the First Earthquake

The first earthquake has a magnitude of 4.2, so we can express its relationship to intensity as \( 4.2 = \log_{10}(I_1/I_0) \), where \( I_1 \) is the intensity of the first earthquake.

Step 3: Calculate Intensity of the First Earthquake

Rearranging the equation from Step 2, we get \( I_1 = I_0 \times 10^{4.2} \). This gives the intensity of the first earthquake in terms of the reference intensity \( I_0 \).

Step 4: Set Up the Equation for the Second Earthquake

The second earthquake has an intensity 20 times greater than the first, so \( I_2 = 20 \times I_1 \). Substitute \( I_1 \) from Step 3: \( I_2 = 20 \times I_0 \times 10^{4.2} \).

Step 5: Calculate Magnitude of the Second Earthquake

The magnitude \( M_2 \) of the second earthquake is given by \( M_2 = \log_{10}(I_2/I_0) = \log_{10}(20 \times I_0 \times 10^{4.2}/I_0) \). Simplify to get \( M_2 = \log_{10}(20 \times 10^{4.2}) \).

Step 6: Simplify and Solve the Equation

Further simplify using the properties of logarithms: \( M_2 = \log_{10}(20) + \log_{10}(10^{4.2}) = \log_{10}(20) + 4.2 \). Compute \( \log_{10}(20) \), which is approximately 1.301. Therefore, \( M_2 = 1.301 + 4.2 \).

Step 7: Final Calculation

Add the values from Step 6: \( M_2 = 5.501 \).

Key Concepts

Understanding Earthquake Intensity

When we talk about the intensity of an earthquake, we're referring to the measure of the size or strength of the earthquake itself. This is different from magnitude, which quantifies the energy released by an earthquake. Intensity helps describe how much the ground shakes, the sound of the earthquake, and its visible impact. It's an observational measure and can vary based on location.

• The same earthquake may exhibit different levels of intensity in different areas.
• Intensity is often measured using the Modified Mercalli Intensity (MMI) scale.
• It's important to understand that intensity is a subjective measure.

In contrast, magnitude is calculated using standardized data from seismographs and expressed as a single number on scales like the Richter scale. Intensity helps us understand the localized experience of an earthquake.

Exploring the Logarithmic Scale

The Richter scale is not linear but logarithmic. This means each increment on the scale represents a tenfold increase in measured amplitude. Put simply:

• An earthquake that measures 5 on the Richter scale isn't just "one more" than one measuring 4, it's 10 times more powerful in terms of ground motion.
• If here is a two unit increase, that means a hundredfold increase in amplitude.
• The energy release of an earthquake goes up about 31.6 times with each whole number increase on the scale.

This logarithmic nature allows the Richter scale to cover a broad range of earthquake sizes, from small tremors to large, devastating events. This helps scientists assess and compare earthquakes easily using a common scale.

Performing Magnitude Calculation

Magnitude calculation on the Richter scale involves some understanding of logarithms. To calculate magnitude, you'd use the formula: \[ M = \log_{10}(I/I_0) \]where:

• \( M \) is the magnitude, representing the earthquake's energy.
• \( I \) is the intensity or amplitude of the earthquake.
• \( I_0 \) is a standard reference intensity.

In our example, we started with an earthquake that had a magnitude of 4.2 and an intensity noted as \( I_1 \). We then calculated the intensity for an earthquake with 20 times the physical ground motion using: \[ I_2 = 20 \times I_1 \]Finally, the magnitude of this new earthquake was derived from logarithmic functions, showing the increase to a magnitude of 5.501 once calculations were complete. This process highlights the elegance and utility of logarithmic scales for expressing large variations efficiently.