Problem 46 - Essential Calculus

Question

On the Richter scale, the magnitude \(R\) of an earthquake of intensity \(I\) is \(R=\frac{\ln I-\ln I_{0}}{\ln 10}\) where \(I_{0}\) is the minimum intensity used for comparison. Assume that \(I_{0}=1\) (a) Find the intensity of the 1906 San Francisco earthquake \((R=8.3)\) (b) Find the factor by which the intensity is increased if the Richter scale measurement is doubled. (c) Find \(d R / d I\).

Step-by-Step Solution

Answer

The intensity associated with a \(R = 8.3\) earthquake on the Richter scale is \(I = 10^{8.3}\). The intensity is increased by a factor of \(10^{8.3}\) if the Richter scale measurement is doubled. The derivative of the Richter scale with respect to Intensity is given by \(d R / d I = \frac{1}{I \cdot \ln 10}\)

Step 1: Calculating the intensity of the earthquake

Substitute \(R = 8.3\) and \(I_{0} = 1\) in the formula \(R=\frac{\ln I-\ln I_{0}}{\ln 10}\) to calculate the intensity. The equation becomes\(8.3=\frac{\ln I-\ln 1}{\ln 10}\). Now, solve for I. So, \(I = 10^{8.3}\)

Step 2: Calculation of intensity change

To find how much the intensity increases when the Richter scale measurement is doubled, substitute \(R = 2R\) into the formula and find the new \(I'\). \(2R = \frac{\ln I' - \ln I_{0}}{\ln 10}\), which simplifies to \(2 \cdot 8.3 = \frac{\ln I'}{\ln 10}\). Solving for \(I'\) gives \(I' = 10^{2 \cdot 8.3}\). The factor by which intensity is increased is then \(I' / I = 10^{2 \cdot 8.3} / 10^{8.3} = 10^{8.3}\)

Step 3: Finding the derivative of Richter scale with respect to intensity

The derivative of R with respect to I, \(dR/dI\), should be found using logarithm differentiation rules. Starting from the base relation \(R=\frac{\ln I - \ln I_{0}}{\ln 10}\). Because \(\ln I_{0}\) is a constant, its derivative will be 0. Therefore, \(dR / dI = \frac{1}{I \cdot \ln 10}\)

Key Concepts

Earthquake Intensity

Earthquake intensity is a measure of how much energy an earthquake releases. Scientists use the Richter Scale to quantify this intensity. The equation \( R = \frac{\ln I - \ln I_{0}}{\ln 10} \) explains the relationship between the Richter magnitude \( R \) and the intensity \( I \). Here, \( I_{0} \) represents the minimum threshold of intensity used for comparison, which is often set to 1 for simplification.
For example, to find the intensity of the infamous 1906 San Francisco earthquake with a magnitude of 8.3, we rearrange the formula:

• Substitute \( R = 8.3 \) and \( I_{0} = 1 \) into the formula.
• Solve for \( I \) to obtain \( I = 10^{8.3} \).

This means the intensity of that earthquake was \( 10^{8.3} \) times the base threshold of intensity. This logarithmic scale helps compare very different energy releases in a meaningful way.

Differentiation in Calculus

Differentiation in calculus is a process used to determine how a function changes as its input changes. For the Richter Scale, finding \( \frac{dR}{dI} \) allows us to understand how sensitive the Richter number is to changes in intensity.
The equation is: \( R = \frac{\ln I - \ln I_{0}}{\ln 10} \). Differentiating with respect to \( I \) involves these steps:

• Treat \( \ln I_{0} \) as a constant since it represents a fixed baseline, its derivative is zero.
• Apply logarithmic differentiation rules: the derivative of \( \ln I \) is \( \frac{1}{I} \).
• Thus, we get \( \frac{dR}{dI} = \frac{1}{I \cdot \ln 10} \).

This result reveals that as intensity \( I \) increases, the rate of change of the Richter number with respect to intensity decreases, indicating a logarithmic relationship.

Logarithmic Functions

Logarithmic functions help simplify equations involving exponents and are foundational in many scientific calculations. The Richter Scale itself is a logarithmic measure. It uses the natural logarithm \( \ln \), which is based on the mathematical constant \( e \). This makes it ideal for representing earthquake intensities, which can vary greatly across magnitudes.
The key features of logarithms include:

• Converting multiplicative processes into additive ones: \( \ln(ab) = \ln a + \ln b \).
• Handling big numbers like intensities compactly.
• Inverting exponential expressions: if \( I = e^x \), then \( x = \ln I \).

In the context of the earthquake problem, the initial formula \( R = \frac{\ln I - \ln I_{0}}{\ln 10} \) elegantly illustrates how small changes on a logarithmic scale translate into exponential changes in real-world terms. This is why a small increase in the Richter number can represent a dramatic increase in energy release.