Problem 68 - Calculus for Biology and Medicine

Question

Rundle et al. (2003) showed that earthquakes in Southern California obey an exponential distribution-that is, if \(N(m)\) is the number of earthquakes in a given year whose magnitude exceeds \(m\), then $$ N(m)=c \cdot 10^{-m} $$ where \(c\) is a positive constant. (a) Suppose in a given year there are 10 earthquakes of magnitude 5 or above. (i) Calculate the constant \(c\). (ii) How many earthquakes will have magnitudes exceeding 2 ? ( 2 is the threshold at which earthquakes can be felt by most people.) (iii) How many earthquakes will have magnitude exceeding 6 ? (6 is the threshold for an earthquake to be regarded as strong.)

Step-by-Step Solution

Answer

\( c = 10^6 \); 10,000 earthquakes exceed magnitude 2; 10 exceed magnitude 6.

Step 1: Understand the Given Information

We have the formula \( N(m) = c \cdot 10^{-m} \) where \( N(m) \) is the number of earthquakes with magnitude exceeding \( m \). We are given that in a certain year, there are 10 earthquakes with magnitudes 5 or above.

Step 2: Calculate the Constant c

We know \( N(5) = 10 \). Substituting in the formula gives \( 10 = c \cdot 10^{-5} \). Solving for \( c \), we rearrange to find \( c = 10^6 \).

Step 3: Calculate Earthquakes with Magnitude Exceeding 2

Using \( c = 10^6 \), substitute into the formula for \( m = 2 \): \( N(2) = 10^6 \cdot 10^{-2} = 10^4 \). Therefore, there are 10,000 earthquakes with magnitudes greater than 2.

Step 4: Calculate Earthquakes with Magnitude Exceeding 6

Substitute \( m = 6 \) into the formula with \( c = 10^6 \): \( N(6) = 10^6 \cdot 10^{-6} = 10 \). Thus, there are 10 earthquakes with magnitudes exceeding 6.

Key Concepts

Earthquake Magnitude

Earthquake magnitude is a key concept when discussing seismic activities. It describes the size or amplitude of an earthquake. The magnitude is usually measured using the Richter scale or similar methods that quantify the energy released. In the context of the exercise, understanding the magnitude is crucial because it helps determine how many earthquakes fall into a certain category of strength. Richter Scale Basics

• The Richter scale is logarithmic. A one-unit increase means a tenfold increase in the amplitude of the seismic waves.
• A magnitude 2 earthquake is often barely felt, while a magnitude 5 can cause damage.
• The magnitude is calculated based on the amplitude of the waves recorded by seismographs.

For the assignment, the magnitude determines the thresholds used to calculate the number of earthquakes exceeding a particular strength. This helps in understanding how frequent stronger or significant events are.

Constant Calculation

Calculating the constant, often denoted as \( c \), requires understanding the relationship between variables in the exponential formula provided. This constant adjusts the formula to fit observed data, such as the number of earthquakes above a certain magnitude.Finding the Constant

• The formula given is \( N(m) = c \cdot 10^{-m} \).
• When the magnitude threshold and corresponding number of earthquakes are known (e.g., \( N(5) = 10 \)), you can solve for \( c \).

In the example, if there are 10 earthquakes of magnitude at least 5, then \( 10 = c \cdot 10^{-5} \). Solving for \( c \) gives \( c = 10^6 \). This constant \( c \) then applies to the entire region or dataset, allowing you to compute expected earthquake frequencies for other magnitudes.

Magnitude Threshold

The magnitude threshold refers to specific points on the magnitude scale that separate one category of earthquakes from another. They are crucial for understanding and predicting the number of earthquakes exceeding a certain intensity. Examples in the Exercise

• Magnitude 2: This is generally the threshold at which earthquakes can begin to be felt by humans. The exercise uses this to calculate how many such noticeable events occur.
• Magnitude 6: This indicates a strong earthquake capable of causing significant damage, prompting the calculation of events at or above this magnitude.

Identifying these thresholds helps both scientists and the public prepare for the potential impact of seismic activities by assessing the frequency and severity of such events.

Seismic Distribution

Seismic distribution describes how earthquakes are spread across different magnitudes in a specific region. This distribution often follows patterns or laws, such as the Gutenberg-Richter law, which helps in analyzing earthquake likelihood and impact.Understanding Exponential Distribution

• The distribution in the exercise follows the form \( N(m) = c \cdot 10^{-m} \), showing a decline in the number of earthquakes as magnitude increases.
• An exponential distribution means smaller, less intense earthquakes are more common, while larger ones are rarer.

This type of distribution helps researchers predict the frequency of different magnitude earthquakes, aiding in risk assessment and preparedness efforts. Recognizing this pattern allows planners to allocate resources efficiently to mitigate the effects of potential seismic events.