Problem 77 - College Algebra with Corequisite Support

Question

The magnitude \(M\) of an earthquake is represented by the equation \(M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\) where \(E\) is the amount of energy released by the earthquake in joules and \(E_{0}=10^{4.4}\) is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing \(1.4 \cdot 10^{13}\) joules of energy?

Step-by-Step Solution

Answer

The magnitude is approximately 5.83.

Step 1: Identify the given values and the formula

We are given that the energy released by the earthquake is \(1.4 \cdot 10^{13}\) joules. The formula for the magnitude \(M\) is given by \(M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\), where \(E_{0}=10^{4.4}\).

Step 2: Substitute the known values into the formula

Substitute \(E=1.4 \times 10^{13}\) and \(E_{0}=10^{4.4}\) into the formula. The expression becomes:\[ M = \frac{2}{3} \log \left(\frac{1.4 \times 10^{13}}{10^{4.4}}\right) \]

Step 3: Calculate the expression inside the logarithm

First, divide the energies:\[ \frac{1.4 \times 10^{13}}{10^{4.4}} = 1.4 \times 10^{13-4.4} = 1.4 \times 10^{8.6} \]

Step 4: Evaluate the logarithm

Calculate the logarithm of \(1.4 \times 10^{8.6}\):\[ \log(1.4 \times 10^{8.6}) = \log(1.4) + \log(10^{8.6}) = \log(1.4) + 8.6 \]

Step 5: Calculate the total value of the logarithm

Use the approximate value \( \log(1.4) \approx 0.1461\). Thus:\[ \log(1.4) + 8.6 \approx 0.1461 + 8.6 = 8.7461 \]

Step 6: Determine the magnitude using the calculated logarithm

Substitute the value back into the magnitude formula:\[ M = \frac{2}{3} \times 8.7461 \approx 5.8307 \]

Step 7: Round the magnitude to the nearest hundredth

The calculation gives \(M \approx 5.8307\). Rounding to the nearest hundredth, we find \(M \approx 5.83\).

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential concept in mathematics, particularly useful in real-world applications like calculating the magnitude of earthquakes. When we talk about logarithms, we're referring to the power to which a number—known as the base—must be raised to produce a certain value. For example, the common logarithm uses base 10.

Logarithmic functions are often expressed as "\(\log_{b}(x)\)," where \(b\) is the base and \(x\) is the value we're considering. In the context of earthquakes, logarithms help us manage the vast differences in energy release. They simplify complex calculations, making massive or minuscule values easier to work with.

Some key points to remember regarding logarithmic functions include:

• The change of base formula which allows us to rewrite logarithms with a different base.
• Logarithmic properties such as \(\log(ab) = \log a + \log b\) and \(\log(a^b) = b \cdot \log a\).

Understanding logarithms is crucial for interpreting how energy released during an earthquake can be translated into its magnitude on the Richter scale.

Energy in Earthquakes

The energy of an earthquake is a pivotal factor in determining its magnitude and therefore its potential impact. Energy in earthquakes is typically measured in joules, a unit of measurement in physics that quantifies work, energy, or heat.In our exercise, the energy released by the earthquake is given as \(1.4 \times 10^{13}\) joules, which is a significant amount, indicating a strong earthquake.

In seismology, energy measurements help in quantifying and understanding the earthquake’s impact. It informs us about:

• How much ground shaking occurred.
• The potential for damage to infrastructure.
• How far-reaching the effects might be.

To provide a baseline, a minimal reference energy, \(E_0\), is used in calculations, assigned as \(10^{4.4}\) joules in our case. This baseline allows for a standardized method for scientists to compare different earthquakes consistently.

Mathematical Formulas

Mathematical formulas are the backbone of solving seismological problems like calculating earthquake magnitudes. A formula like \(M = \frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\) is built upon principles of physics and mathematics to yield reliable and interpretable results.

Understanding each component is crucial:

• \(M\), representing the magnitude, a dimensionless number that gives an idea of the earthquake's intensity.
• \(\frac{2}{3}\), a scaling factor adjusting the logarithm to fit within a determined range for magnitudes.
• \(E\), the energy produced by the quake, central to the calculation.
• \(E_{0}\), the reference energy used as a benchmark for other energy measurements.

Taking mathematical calculations and converting them into practical information allows us to understand natural phenomena such as earthquakes better.By substituting the given values into the formula and simplifying, we find the solution to our exercise.

Scientific Calculations

Scientific calculations often involve complex steps, requiring a deep understanding of mathematical principles and attention to detail. In the given earthquake magnitude problem, the calculations focus on evaluating and interpreting scientific data through mathematical operations.

The steps to solve such problems include:

• Identifying the known values from the question.
• Rearranging the known values into the mathematical formula.
• Simplifying the expression, including operations like division and exponent calculations.
• Performing logarithmic evaluations, using properties like \(\log(1.4 \times 10^{8.6}) = \log(1.4) + 8.6\).
• Calculating results to the desired precision, in this case rounding the magnitude to the nearest hundredth.

By understanding these calculations, you gain more than just a number; you gain insights into how the Earth's energies transform and impact our world.Scientific calculations serve as a bridge between theoretical mathematics and tangible real-world outcomes.