Problem 45 - Calculus and its applications

Question

The magnitude \(\mathrm{R}\) (measured on the Richter scale) of an earthquake of intensity I is defined as \(R=\log \frac{I}{I_{0}},\) where \(I_{0}\) is a minimum intensity used for comparison. If one earthquake is 10 times as intense as another, its magnitude on the Richter scale is 1 higher; if one earthquake is 100 times as intense as another, its magnitude is 2 higher, and so on. Thus, an earthquake whose magnitude is 6 on the Richter scale is 10 times as intense as an earthquake whose magnitude is \(5,\) and 100 times as intense as an earthquake whose magnitude is 4 Use this information. On January \(12,2010,\) a devastating earthquake struck the Caribbean nation of Haiti. It had an intensity that was \(10,000,000,\) or \(10^{7},\) times as intense as \(I_{0}\). What was this earthquake's magnitude on the Richter scale? (Hint: Let \(\left.I=10^{7} \cdot I_{0} .\right)\)

Step-by-Step Solution

Answer

The earthquake's magnitude was 7 on the Richter scale.

Step 1: Define the Magnitude Formula

The magnitude of an earthquake on the Richter scale is defined by the formula \( R = \log \left( \frac{I}{I_0} \right) \). Here, \( I \) is the intensity of the earthquake, and \( I_0 \) is a comparative minimum intensity.

Step 2: Substitute the Given Intensity

We are given that the earthquake's intensity is \( I = 10^7 \cdot I_0 \). Substitute this into the formula: \[ R = \log \left( \frac{10^7 \cdot I_0}{I_0} \right) \].

Step 3: Simplify the Expression

The expression simplifies by canceling out \( I_0 \) in the numerator and the denominator: \[ R = \log (10^7) \].

Step 4: Apply Logarithm Properties

The logarithm property states \( \log (a^b) = b \cdot \log(a) \). Apply this to \( \log (10^7) \): \[ R = 7 \cdot \log(10) \].

Step 5: Use Known Logarithm Value

The base 10 logarithm of 10 is \( \log(10) = 1 \). Substitute this value: \[ R = 7 \cdot 1 \].

Step 6: Calculate the Richter Magnitude

Thus, the magnitude \( R \) of the earthquake is \( 7 \).

Key Concepts

Logarithms

Logarithms are a crucial mathematical tool that help in understanding exponential relationships. In the context of earthquakes, log functions are used to express the magnitude of earthquake intensity levels on the Richter scale. The basic principle of logarithms is based on finding the power to which a given base number must be raised to produce a specific number.

The expression \( \log(b) = x \) translates to "b is the base raised to the power of x equals the number we are considering." For example, when dealing with base 10 (the common logarithm), if \( \log(100) = x \), then 10 raised to the power of x is 100. In simpler terms:

• Base "10"^x = "number"
• Logarithms determine this power x

This makes logarithms an excellent tool for analyzing phenomena with a wide range of values, such as the intensity of natural disasters like earthquakes.

Earthquake Intensity

Earthquake intensity measures how much energy is released at the source of the earthquake. It reflects the actual effect of the earthquake's energy, which can vary greatly from one earthquake to another.

In the Richter scale formula, intensity is denoted by \( I \) and is compared against a standard reference intensity, \( I_0 \). The ratio of these two intensities provides a relative measure.

• \( I \) is the earthquake's intensity
• \( I_0 \) is the reference intensity, representing the smallest detectable seismic activity

The Richter scale helps to categorize earthquakes into a range of magnitudes based on their intensities, aiding in understanding potential damage and necessary precautionary measures.

Magnitude Calculation

Calculating the magnitude of an earthquake on the Richter scale involves using logarithms to relate the intensity of an earthquake to its magnitude. The formula \( R = \log \left( \frac{I}{I_0} \right) \) defines this relationship, where \( R \) is the magnitude and \( \frac{I}{I_0} \) is the intensity ratio.

Let's break it down with an example using the given exercise:

• Suppose the intensity \( I = 10^7 \cdot I_0 \), meaning the earthquake is 10 million times more intense than the reference level
• Plug this intensity into the formula: \( R = \log \left( \frac{10^7 \cdot I_0}{I_0} \right) \)
• Simplifying, this becomes \( R = \log(10^7) = 7 \)
• Thus, the earthquake's magnitude is 7

Understanding these calculations helps in assessing the potential impact of seismic events, with each unit increase on the Richter scale representing a tenfold increase in measured amplitude and roughly 31.6 times more energy release.