Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

The magnitude \(M\) of an earthquake on the Richter scale is $$ M=0.67 \log _{10}(0.37 E)+1.46 $$ where E is the energy of the earthquake in kilowatt-hours. Find the energy of an earthquake of magnitude 7. Of magnitude 8.

Step-by-Step Solution

Answer

Energy for M=7: \(5.51 \times 10^8\) kWh; for M=8: \(3.72 \times 10^{10}\) kWh.

Step 1: Understand the formula

The formula for the magnitude of an earthquake on the Richter scale is given by \( M = 0.67 \log_{10}(0.37E) + 1.46 \). We need to find the energy \( E \) when the magnitude \( M \) is given.

Step 2: Rearrange the formula

To find the energy \( E \), we first need to isolate the logarithmic expression. Thus, rearrange the formula to get \( \log_{10}(0.37E) = \frac{M - 1.46}{0.67} \).

Step 3: Solve for E when M=7

Substitute \( M = 7 \) into the rearranged formula to find \( E \):\[\log_{10}(0.37E) = \frac{7 - 1.46}{0.67} = \frac{5.54}{0.67} \approx 8.27\]Convert this to exponential form: \( 0.37E = 10^{8.27} \).Solve for \( E \):\[E = \frac{10^{8.27}}{0.37}\]

Step 4: Calculate the energy for M=7

Calculate \( E \) using the expression found:\[E = \frac{10^{8.27}}{0.37} \approx 5.51 \times 10^8 \text{ kilowatt-hours}\]

Step 5: Solve for E when M=8

Now substitute \( M = 8 \) into the rearranged formula:\[\log_{10}(0.37E) = \frac{8 - 1.46}{0.67} = \frac{6.54}{0.67} \approx 9.75\]Convert this to exponential form: \( 0.37E = 10^{9.75} \).Solve for \( E \):\[E = \frac{10^{9.75}}{0.37}\]

Step 6: Calculate the energy for M=8

Calculate \( E \) using this expression:\[E = \frac{10^{9.75}}{0.37} \approx 3.72 \times 10^{10} \text{ kilowatt-hours}\]

Key Concepts

Earthquake Magnitude

When discussing earthquakes, the term "magnitude" refers to a measure of the size or energy release of an earthquake. The Richter scale is a common scale used for this purpose, which quantifies the magnitude as a logarithmic measure. Every increase of one unit on the Richter scale corresponds to a tenfold increase in amplitude of the seismic waves as recorded by seismographs. Furthermore, it represents approximately 31 times more energy release. This relation highlights why earthquakes of higher magnitudes are exponentially more destructive. The Richter scale allows scientists to easily compare the size of different earthquakes.

Logarithmic Equation

The Richter scale equation uses logarithms, a mathematical tool that helps deal with the incredibly large range of values earthquake energies can have. In the formula given, the magnitude \( M \) is calculated from the energy \( E \) as follows:

• \( M = 0.67 \log_{10}(0.37E) + 1.46 \)

This equation involves a base 10 logarithm, which expresses energy \( E \) in kilowatt-hours. The logarithmic component \( \log_{10}(0.37E) \) allows us to tackle the wide potential values of \( E \), converting them into a manageable scale. Logarithms essentially compress the data so that extremely high values of energy can be handled in a simpler form on the Richter scale.

Energy Calculation

To find the energy \( E \) from a given magnitude \( M \) on the Richter scale, you need to reverse the logarithmic equation process. Start by isolating the logarithmic term:

• \( \log_{10}(0.37E) = \frac{M - 1.46}{0.67} \)

Then, convert the logarithm to its exponential form to solve for \( E \):

• \( 0.37E = 10^{\frac{M - 1.46}{0.67}} \)
• \( E = \frac{10^{\frac{M - 1.46}{0.67}}}{0.37} \)

This conversion allows you to calculate \( E \) for any given magnitude, by using the inverse property of logarithms and exponents. Practically multiplying and dividing by coefficients maintains the integrity of the original equation, leading to a direct calculation of seismic energy in kilowatt-hours for a specific earthquake magnitude.

Seismic Energy

Seismic energy is the total energy released by an earthquake. Calculating the seismic energy, as we have seen, involves more than simply looking at the severity felt on the surface. The expression for \( E \), derived from the logarithmic formula, gives us a specific numeric value describing the amount of energy an earthquake releases in kilowatt-hours. For instance, calculating \( E \) for a magnitude \( 7 \) earthquake reveals an energy level of approximately \( 5.51 \times 10^8 \) kilowatt-hours, while a magnitude \( 8 \) earthquake results in a dramatically higher \( 3.72 \times 10^{10} \) kilowatt-hours. These figures underscore the substantial increase in energy release with each additional magnitude, highlighting the critical need for understanding and accurately computing seismic energy to prepare for potential impacts.