Problem 66 - College Algebra with Corequisite Support

Question

The intensity levels \(I\) of two earthquakes measured on a seismograph can be compared by the formula \(\log \frac{I_{1}}{I_{2}}=M_{1}-M_{2}\) where \(M\) is the magnitude given by the Richter Scale. In August \(2009, \mathrm{an}\) earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of \(9.0 .{ }^{8}\) How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

Step-by-Step Solution

Answer

The 2011 earthquake was approximately 794 times greater in intensity than the 2009 earthquake.

Step 1: Identify Known Values

We know from the problem that the magnitude of the first earthquake, \( M_1 \), is 6.1 and the magnitude of the second earthquake, \( M_2 \), is 9.0.

Step 2: Set Up the Formula

The formula given is \( \log \left( \frac{I_1}{I_2} \right) = M_1 - M_2 \). We will substitute the known values: \( \log \left( \frac{I_1}{I_2} \right) = 6.1 - 9.0 \).

Step 3: Calculate the Magnitude Difference

Subtract the magnitudes: \( 6.1 - 9.0 = -2.9 \). Thus, \( \log \left( \frac{I_1}{I_2} \right) = -2.9 \).

Step 4: Convert Logarithmic Equation to Exponential Form

The equation \( \log \left( \frac{I_1}{I_2} \right) = -2.9 \) is equivalent to \( \frac{I_1}{I_2} = 10^{-2.9} \) in exponential form.

Step 5: Solve for Intensity Ratio

Calculate \( 10^{-(-2.9)} = 10^{2.9} \) to find \( \frac{I_1}{I_2} \).

Step 6: Compute the Intensity Ratio

Calculate \( 10^{2.9} \) which is approximately 794.33.

Step 7: Round to the Nearest Whole Number

Round 794.33 to the nearest whole number, which is 794.

Key Concepts

Magnitude

The magnitude of an earthquake refers to the size or strength of the seismic event. It's a numerical value that represents the amount of energy released by an earthquake. One commonly used scale for this purpose is the Richter Scale. Magnitude is crucial in determining the impact of an earthquake, as higher magnitudes typically indicate stronger earthquakes capable of causing more damage.

The Richter Scale provides a logarithmic representation of the earthquake's size. For instance, an earthquake of magnitude 9.0 releases significantly more energy than one of magnitude 6.1. This is because each whole number increase on the Richter Scale corresponds to a tenfold increase in measured amplitude and roughly 31.6 times more energy release.

Thus, understanding magnitude helps us compare the relative energies of different earthquakes.

Seismograph

A seismograph is an instrument that measures and records the details of seismic waves generated by earthquakes. It detects and measures the movement of the ground, making it crucial for earthquake analysis.

Seismographs work by using a suspended mass, which remains still while the rest of the device moves with the Earth’s motion during an earthquake. The relative motion between the mass and the frame is converted into an electrical signal that can be recorded digitally or on paper.

Seismographs provide essential data that allow seismologists to calculate the magnitude and locate the epicenter of an earthquake, aiding in the assessment of potentially affected areas.

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms, which are the inverse operations of exponentiation. Simply put, if you know the logarithmic value, you can find the original number that was raised to a power, or exponent, to get it. In our context, logarithmic equations are used on the Richter Scale to calculate the magnitude of earthquakes.

The formula \( \log \left( \frac{I_1}{I_2} \right) = M_1 - M_2\) allows us to compare the intensity of two earthquakes using their magnitudes. Here, \(M\) represents magnitude, and \(I\) is the intensity.

This equation shows that the difference in magnitude is equal to the logarithm of the ratio of their intensities. By rearranging the equation, we can solve for the intensity ratio, providing insights into just how much stronger one earthquake is compared to another.

Intensity Ratio

The intensity ratio is a way to compare the strength of two earthquake events by examining their respective energy releases. This is done using the equation mentioned earlier: \( \log \left( \frac{I_1}{I_2} \right) = M_1 - M_2\).

The process consists of substituting known magnitudes into the equation, solving for the logarithm of the intensity ratio, and then converting this logarithmic term to an exponential form to find the actual ratio.

For example, if \(M_1 = 6.1\) and \(M_2 = 9.0\), as in our exercise, the difference in magnitudes \(M_1 - M_2\) becomes -2.9. Thus, \(\log \left( \frac{I_1}{I_2} \right) = -2.9\) translates to \(\frac{I_1}{I_2} = 10^{2.9}\), which calculates to approximately 794. This means the intensity of the 2011 earthquake was 794 times greater than that of the 2009 earthquake.

This method allows for a precise numerical comparison, highlighting the, often substantial, difference in the energy released.