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Chapter 4: Problem 59

Use the Richter scale formula \(R=\log \left(I / I_{0}\right)\) to find the magnitude of an earthquake that has an intensity. (a) 100 times that of \(I_{0}\) (b) \(10,000\) times that of \(I_{0}\) (c) \(100,000\) times that of \(I_{0}\)

Short Answer

Expert verified
(a) 2; (b) 4; (c) 5

Step by step solution

01

Understand the Formula

The Richter scale formula is given by \( R = \log \left( \frac{I}{I_0} \right) \), where \( R \) is the magnitude of the earthquake, \( I \) is the intensity of the earthquake, and \( I_0 \) is the reference intensity. We need to compute \( R \) for different values of \( I \).
02

Calculate for Intensity 100 times \( I_0 \)

For \( I = 100 \times I_0 \), substitute into the formula: \( R = \log \left( \frac{100 \times I_0}{I_0} \right) = \log(100) \). Simplifying this, we find \( R = 2 \) because \( \log(100) = \log(10^2) = 2 \cdot \log(10) = 2 \).
03

Calculate for Intensity 10,000 times \( I_0 \)

For \( I = 10,000 \times I_0 \), substitute into the formula: \( R = \log \left( \frac{10,000 \times I_0}{I_0} \right) = \log(10,000) \). Simplifying this, we find \( R = 4 \) because \( \log(10,000) = \log(10^4) = 4 \cdot \log(10) = 4 \).
04

Calculate for Intensity 100,000 times \( I_0 \)

For \( I = 100,000 \times I_0 \), substitute into the formula: \( R = \log \left( \frac{100,000 \times I_0}{I_0} \right) = \log(100,000) \). Simplifying this, we find \( R = 5 \) because \( \log(100,000) = \log(10^5) = 5 \cdot \log(10) = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Earthquake Intensity
Earthquake intensity refers to the amount of energy released by an earthquake. It is a crucial measure that helps scientists and citizens gauge the potential effects and damage caused by the seismic event. The intensity of an earthquake is noted by the symbol \( I \), which quantifies the energy in relation to a standard reference intensity \( I_0 \).

The reference intensity \( I_0 \) is set as a baseline to compare other intensities, thereby allowing a standardized measurement method. Understanding the intensity of an earthquake can help in preparing and responding to seismic activity, which can vary greatly depending on its geographical location and depth.
Logarithmic Function
A logarithmic function is a mathematical function that helps in describing phenomena like growth rates. It is the inverse operation of exponentiation and is expressed as \( \log \). In the context of earthquake magnitudes on the Richter scale, the logarithmic function is employed to translate intensity values into a manageable and understandable scale.

The Richter scale uses a base-10 logarithm, which means each unit increase in magnitude represents an intensity that is ten times greater. For instance, an earthquake with a magnitude of 5 is ten times more intense than one with a magnitude of 4. This logarithmic approach is essential because it provides a clear, concise representation of the vast range of earthquake intensities.
Magnitude Calculation
Magnitude calculation on the Richter scale involves determining earthquake strength based on the formula \( R = \log \left( \frac{I}{I_0} \right) \), where \( R \) is the earthquake's magnitude. This calculation helps in transforming the earthquake's raw intensity into a value on the Richter scale.

  • For an intensity 100 times \( I_0 \), the magnitude \( R \) is calculated as 2 (since \( \log(100) = 2 \)).
  • For an intensity 10,000 times \( I_0 \), the magnitude \( R \) is 4 (since \( \log(10,000) = 4 \)).
  • For an intensity 100,000 times \( I_0 \), the magnitude \( R \) is 5 (since \( \log(100,000) = 5 \)).
These calculations demonstrate the exponential nature of the scale and highlight the difference in energy release associated with each unit increase in magnitude.
Intensity Ratio
The intensity ratio is a comparison of the earthquake's intensity \( I \) to the reference intensity \( I_0 \). Calculating this ratio is integral to determining the magnitude of the seismic activity. The formula \( \frac{I}{I_0} \) expresses how much more intense an earthquake is compared to the baseline intensity.

This ratio is instrumental because it:
  • Provides a clear, standardized way to convey the severity of an earthquake.
  • Allows for comparisons between earthquakes of different sizes and magnitudes.
  • Helps in predicting potential impacts and preparing for future seismic events.
Through the logarithmic scale, an intensity ratio is efficiently converted into a Richter magnitude, offering an accessible means of understanding earthquake data.

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Most popular questions from this chapter

The atmospheric density at altitude \(x\) is listed in the table. $$\begin{array}{lrrr}\hline \text { Altitude }(\mathrm{m}) & 0 & 2000 & 4000 \\\\\hline \text { Density }\left(\mathrm{kg} / \mathrm{m}^{3}\right) & 1.225 & 1.007 & 0.819 \\\\\hline\end{array}$$ $$\begin{array}{|lccc|}\hline \text { Altitude (m) } & 6000 & 8000 & 10,000 \\\\\hline \text { Density (kg/m }^{3} \text { ) } & 0.660 & 0.526 & 0.414 \\\\\hline\end{array}$$ (a) Find a function \(f(x)=C_{0} e^{t x}\) that approximates the density at altitude \(x,\) where \(C_{0}\) and \(k\) are constants. Plot the data and \(f\) on the same coordinate axes. (b)Use \(f\) to predict the density at 3000 and 9000 meters. Compare the predictions to the actual values of 0.909 and \(0.467,\) respectively.

The number of bacteria in a certain culture increased from 600 to 1800 between 7: 00 A.M. and 9: 00 A.M. Assuming growth is exponential, the number \(f(t)\) of bacteria \(t\) hours after 7: 00 A.M. is given by \(f(t)=600(3)^{1 / 2}\) (a) Estimate the number of bacteria in the culture at 8: 00 A.M., 10: 00 A.M., and 11: 00 A.M. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 4\)

Children's weight The Ehrenberg relation $$\ln W=\ln 2.4+(1.84) h$$ is an empirically based formula relating the height \(h\) (in meters) to the average weight \(W\) (in kilograms) for children 5 through 13 years old. (a) Express \(W\) as a function of \(h\) that does not contain In. (b) Estimate the average weight of an 8 -year-old child who is 1.5 meters tall.

The loudness of a sound, as experienced by the human ear, is based on its intensity level. A formula used for finding the intensity level \(\alpha\) (in decibels) that corresponds to a sound intensity \(I\) is \(\alpha=10 \log \left(I / I_{0}\right),\) where \(I_{0}\) is a special value of \(I\) agreed to be the weakest sound that can be detected by the ear under certain conditions. Find \(\alpha\) if. (a) \(I\) is 10 times as great as \(I_{0}\) (b) \(I\) is 1000 times as great as \(I_{0}\) (c) \(I\) is \(10,000\) times as great as \(I_{0}\) (This is the intensity level of the average voice.)

If monthly payments \(p\) are deposited in a savings account paying an annual interest rate \(r,\) then the amount \(A\) in the account after \(n\) years is given by$$A=\frac{p\left(1+\frac{r}{12}\right)\left[\left(1+\frac{r}{12}\right)^{12 n}-1\right]}{\frac{r}{12}}$$Graph \(A\) for each value of \(p\) and \(r,\) and estimate \(n\) for \(A=\$ 100,000\) $$p=250, \quad r=0.09$$
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