Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Use the Richter scale formula \(R=\log \left(I / I_{0}\right)\) to find the magnitude of an earthquake that has an intensity (a) 100 times that of \(I_{0}\) (b) 10,000 times that of \(I_{0}\) (c) 100,000 times that of \(I_{0}\)

Step-by-Step Solution

Answer

The magnitudes are 2, 4, and 5.

Step 1: Understanding the Richter Scale Formula

The Richter scale formula is given as \( R = \log \left( \frac{I}{I_0} \right) \), where \( R \) is the Richter magnitude, \( I \) is the intensity of the earthquake, and \( I_0 \) is a reference intensity. The formula calculates the magnitude based on the logarithm of the ratio of the earthquake's intensity to a standard reference intensity.

Step 2: Calculating Magnitude for Intensity 100 Times \( I_0 \)

Let \( I = 100 \times I_0 \). Substituting \( I \) into the formula, we have:\[R = \log \left( \frac{100I_0}{I_0} \right) = \log (100)\]Since \( \log (100) \) can be expanded as \( \log (10^2) = 2 \times \log (10) = 2 \) (since \( \log (10) = 1 \)), the magnitude \( R = 2 \).

Step 3: Calculating Magnitude for Intensity 10,000 Times \( I_0 \)

Let \( I = 10,000 \times I_0 \). Substituting \( I \) into the formula, we have:\[R = \log \left( \frac{10,000I_0}{I_0} \right) = \log (10,000)\]\( \log (10,000) \) can be simplified as \( \log (10^4) = 4 \times \log (10) = 4 \). Therefore, the magnitude \( R = 4 \).

Step 4: Calculating Magnitude for Intensity 100,000 Times \( I_0 \)

Let \( I = 100,000 \times I_0 \). Substituting \( I \) into the formula, we have:\[R = \log \left( \frac{100,000I_0}{I_0} \right) = \log (100,000)\]\( \log (100,000) \) can be expanded as \( \log (10^5) = 5 \times \log (10) = 5 \). Hence, the magnitude \( R = 5 \).

Key Concepts

Logarithmic Calculations

Logarithmic calculations are essential in the Richter scale formula. In mathematics, a logarithm is the inverse operation to exponentiation. It helps determine how many times one number must be multiplied to get another. This is expressed as \(\log_b(a) = c\) where \(b^c = a\). Such calculations simplify understanding large scales or wide-ranging values.
The base 10 logarithm (common logarithm) is often used because it aligns naturally with our decimal system, which makes it practical for scientific calculations. As seen in the exercise, \(\log(100)\) simplifies to 2 because \(10^2 = 100\). This makes computations easier, especially dealing with exponential growth, like that of earthquakes.
Each step in a logarithmic scale, such as the Richter scale, indicates a tenfold increase in measured intensity, which is pivotal in interpreting earthquake magnitudes.

Earthquake Magnitude

Earthquake magnitude measures the size or energy release of an earthquake. It's crucial for understanding the potential damage and impact. The Richter scale, developed by Charles F. Richter, quantifies this by comparing an earthquake's intensity to a known standard, \(I_0\).
The magnitude \(R\) on the Richter scale is calculated using the formula: \( R = \log\left( \frac{I}{I_0} \right) \). Magnitudes can vary massively, so using a scale is essential for practical comparisons. For instance, a magnitude increase of 1 reflects a tenfold increase in intensity.
Understanding earthquake magnitudes helps scientists and engineers design structures better suited to withstand seismic forces, contributing to improved safety and preparedness.

Intensity Ratio

In the Richter scale formula, the intensity ratio is the comparison of the earthquake's intensity \(I\) to the standard reference intensity \(I_0\). This ratio is inside the logarithmic calculation, providing the basis for determining the earthquake's magnitude.
Understanding this ratio is crucial because it contextualizes the enormous differences in energy released. For example, if an earthquake's intensity is 10,000 times \(I_0\), the magnitude \(R\) becomes \( \log(10,000) = 4 \). Each increase in the magnitude number reflects a tenfold increase in intensity.
The intensity ratio thus allows us to express these large-scale seismic events in a compact, manageable way, making it easier to interpret and communicate the potential severity of earthquakes.

Mathematical Formulas

Mathematical formulas form the backbone of earthquake measurement scales. For the Richter scale, the formula \( R = \log\left( \frac{I}{I_0} \right) \) is a key tool for calculating earthquake magnitudes.
These formulas leverage the properties of logarithms to translate wide-ranging intensity values into manageable numbers. It takes advantage of the fact that a small change in magnitude means a large change in intensity, given the exponential nature of logarithms. For example, a magnitude \(R = 2\) means an intensity 100 times \(I_0\), whereas \(R = 5\) represents an intensity 100,000 times \(I_0\).
Utilizing formulas in this way allows scientists to consistently and accurately describe the powerful forces of nature in a standardized format, aiding in global communication and emergency response.