Problem 69 - Calculus - AP Edition

Question

The magnitude \(R\) of an earthquake on the Richter scale is related to the amplitude \(A\) of the shock wave by the equation $$R=\log _{10}\left(\frac{A}{A_{0}}\right)$$ where \(A_{0}\) is a small positive constant. (Originally, \(A_{0}\) was the smallest possible amplitude that could be detected.) Use differentials to show that for a small change in \(A\) the change in \(R\) can be approximated by multiplying the relative change \(\Delta A / A\) in amplitude by 0.4343.

Step-by-Step Solution

Answer

For a small change in amplitude, the change in Richter magnitude can be approximated by multiplying the relative change in amplitude by 0.4343.

Step 1: Understand the Problem

We are given the equation for the magnitude of an earthquake on the Richter scale: \(R = \log_{10}\left(\frac{A}{A_0}\right)\). We need to use differentials to approximate the change in \(R\) when the amplitude \(A\) changes slightly.

Step 2: Differentiate the Equation

The expression for \(R\) is given as \(R = \log_{10}\left(\frac{A}{A_0}\right)\). We need the derivative \(dR\) with respect to \(A\). Recall that the derivative of \(\log_{10}(x)\) is \(\frac{1}{x \ln(10)}\). Thus, \(dR = \frac{1}{\left(\frac{A}{A_0}\right) \ln(10)} \cdot \frac{1}{A_0} \, dA = \frac{1}{A \ln(10)} \, dA\).

Step 3: Relate dR to the Relative Change in Amplitude

The differential \(dR\) can be expressed in terms of the change in amplitude: \(dR = \frac{dA}{A \ln(10)}\). This shows that the change in \(R\) is proportional to the relative change \(\frac{dA}{A}\) in amplitude.

Step 4: Simplify Using the Constant Approximation

Recognize that \(\ln(10)\) is approximately \(2.3026\). Therefore, \(\frac{1}{\ln(10)} = \frac{1}{2.3026} \approx 0.4343\). We substitute this approximation into our expression: \(dR \approx 0.4343 \times \frac{dA}{A}\).

Step 5: Conclude the Problem

We have shown that for a small change in \(A\), the change in \(R\) can indeed be approximated by multiplying the relative change \(\frac{\Delta A}{A}\) by approximately \(0.4343\).

Key Concepts

Richter Scale

The Richter Scale is a logarithmic scale used to indicate the magnitude of an earthquake. Magnitude is essentially a measure of energy release. This system was created to provide a more accurate representation of an earthquake's size compared to prior methods that lacked precision. The formula used for the Richter scale is \[R = \log_{10}\left(\frac{A}{A_0}\right)\] where:

• R is the Richter magnitude.
• A is the amplitude of the seismic waves.
• A₀ is a reference amplitude, typically the smallest detectable amplitude.

This equation shows how an increase in the amplitude of seismic waves results in an increase in the Richter magnitude. The logarithmic nature of the scale means that each whole number increment represents a tenfold increase in amplitude. This conversion is useful for comparing different earthquakes to understand their relative energy release.

Logarithmic Differentiation

Logarithmic Differentiation is a valuable technique in calculus, particularly when dealing with complicated relationships involving products, quotients, or powers, such as the Richter Scale equation. Here, we're interested in finding the derivative of the function \(R = \log_{10}\left(\frac{A}{A_0}\right)\).
To differentiate a logarithmic function, we use the fact that: \[\frac{d}{dx}\log_{10}(x) = \frac{1}{x \ln(10)}\]
This rule helps us find the differential \(dR\) in terms of \(dA\): \[dR = \frac{1}{\left(\frac{A}{A_0}\right) \ln(10)} \cdot \frac{1}{A_0} \, dA = \frac{1}{A \ln(10)} \, dA\]
This shows the change in the Richter magnitude concerning the change in amplitude. Such differentiation is crucial for understanding how sensitive measures like the Richter scale respond to input changes.

Relative Change

Relative Change describes how much something changes in relation to its original value. In this exercise, we were interested in the relative change of the amplitude \(A\), represented as \(\frac{\Delta A}{A}\). This ratio helps express changes as proportions, which can be more informative than absolute differences.
Through our derivation in the exercise, we used differentials to show the effect of changes in amplitude on the Richter magnitude.
We concluded that: \[dR \approx 0.4343 \times \frac{dA}{A}\] This implies that a small relative change in amplitude translates to a proportionate change in the Richter scale magnitude, multiplied by a factor (approximately 0.4343).
This factor arises from the approximation of \(\ln(10)\) and is a constant that relates logarithmic scale changes to linear relative changes. Understanding relative change and its implications helps in contexts like seismic readings, where understanding the proportionality of change is crucial for accurate predictions and assessments.