Problem 53 - Precalculus with Calculus Previews

Question

(a) Charles Richter working with Beno Gutenberg developed the model $$ M=\frac{2}{3}\left[\log _{10} E-11.8\right] $$ that relates the Richter magnitude \(M\) of an earthquake and its seismic energy \(E\) (measured in ergs). Calculate the seismic energy \(E\) of the 2004 Northern Sumatra earthquake where \(M=9.3\).

Step-by-Step Solution

Answer

The seismic energy \( E \approx 5.62 \times 10^{25} \) ergs.

Step 1: Understand the Given Formula

The formula provided is used to calculate the Richter magnitude \( M \), which is related to the seismic energy \( E \) by \[ M = \frac{2}{3}\left[\log_{10} E - 11.8\right]. \] We need to rearrange this formula to solve for \( E \).

Step 2: Plug in the Given Data

We are given that \( M = 9.3 \) for the 2004 Northern Sumatra earthquake. Therefore, substitute \( M = 9.3 \) into the equation: \[ 9.3 = \frac{2}{3}\left[\log_{10} E - 11.8\right]. \]

Step 3: Isolate the Logarithmic Term

Multiply both sides by \( \frac{3}{2} \) to isolate the logarithmic term: \[ \log_{10} E - 11.8 = \frac{3}{2} \times 9.3. \] Calculate the result: \[ \log_{10} E - 11.8 = 13.95. \]

Step 4: Solve for the Logarithm of Energy

Add 11.8 to both sides to solve for \( \log_{10} E \): \[ \log_{10} E = 13.95 + 11.8 = 25.75. \]

Step 5: Calculate the Seismic Energy

Convert the logarithmic form into exponential form to solve for \( E \): \[ E = 10^{25.75}. \] Calculate the result: \( E \approx 5.62 \times 10^{25} \) ergs.

Key Concepts

Seismic Energy Calculation

The concept of seismic energy is vital in understanding the power behind earthquakes. This energy, often measured in ergs, can be calculated using the Richter magnitude model, developed to quantify an earthquake's intensity. Equipped with the formula \( M = \frac{2}{3}[\log_{10} E - 11.8] \), one can find how much energy was released during an earthquake. Here, \( E \) represents the seismic energy, and the model provides an equation connecting this energy to a measurable scale \( M \). Understanding how to manipulate this formula is crucial, as it allows for solving \( E \) when the magnitude \( M \) is known. By rearranging the equation and isolating the energy term, we can determine the total energy released using the given data about an earthquake's magnitude.

Logarithmic Equations

Logarithmic equations play a crucial role in many scientific calculations, including those related to earthquakes. In the given Richter magnitude formula, \( \log_{10} E \) helps describe how seismic energy scales. Logarithms are used to simplify large numbers, like seismic energy, to a more manageable form.
When working with the formula for seismic energy, one must isolate \( \log_{10} E \). This entails rearranging the equation to solve for the logarithm directly. Multiplying through with fractional coefficients and isolating the logarithmic part makes it easier to proceed.
The manipulation of logs, especially adding or subtracting constants from them, is a foundational mathematical skill. Recognizing how to reverse the logs later is key, helping translate the logarithmic result into its original exponential value. This understanding enables converting logarithmic data back into its original size to comprehend what it truly represents in exponential terms.

Exponential Functions

Exponential functions are fundamental when calculating large-scale phenomena like seismic activities. An exponential function converts a logarithmic result back into an easily understandable number, especially when dealing with data like seismic energy.
In the formula, once \( \log_{10} E = 25.75 \) is achieved, transforming it to \( E = 10^{25.75} \) is essential. This process is known as exponentiation. It allows us to switch from compact logarithmic terms back to raw energy figures, showing the true scale of an event like an earthquake in numbers that can seem vast and abstract.
Exponents can express extremely large or small quantities succinctly. In essence, achieving \( E = 10^{25.75} \) shows the immense energy released during an earthquake in a clear, yet compact form, essential for scientists to evaluate the potential impact or damage of seismic events.