Problem 87 - Calculus for Biology and Medicine

Question

The Richter magnitude scale is used to measure the strength of earthquakes. The magnitude \(m\) of an earthquake is calculated from the amplitude of shaking, \(A\) (measured in \(\mu \mathrm{m}\), where \(1 \mu \mathrm{m}=10^{-6} \mathrm{~m}\) ), measured by a seismometer, and from the distance of the seismometer to the epicenter of the earthquake, \(D\) (measured in \(\mathrm{km}\) ), using the following formula. $$ m=\log A-2.48+2.76 \log D $$ (a) A seismometer distance \(100 \mathrm{~km}\) from the earthquake epicenter measures shaking with an amplitude of \(100 \mu \mathrm{m} .\) Calculate \(m .\) (b) The smallest amplitude of shaking that most people can feel is \(1 \mathrm{~mm}\left(1 \mathrm{~mm}=10^{3} \mu \mathrm{m}\right) .\) Calculate the smallest magnitude of earthquake a person might feel if they were \(10 \mathrm{~km}\) away from the earthquake epicenter. (c) An earthquake is measured to have magnitude \(m=7.2\). Calculate the amplitude of shaking if (i) \(D=10 \mathrm{~km}\) from the epicenter. (ii) \(D=100 \mathrm{~km}\) from the epicenter. (d) Measured at the same distance from the epicenter, an increase of 1 in the Richter magnitude of an earthquake (e.g., from \(m=3\) to \(m=4\) ) corresponds to what factor increase in the amplitude of shaking?

Step-by-Step Solution

Answer

(a) 5.04; (b) 3.28; (c.i) \(8.32 \times 10^6 \, \mu m\); (c.ii) \(1.45 \times 10^4 \, \mu m\); (d) 10 times.

Step 1: Calculate the Magnitude of Earthquake (Part a)

Use the given formula: \( m = \log A - 2.48 + 2.76 \log D \). Substitute \( A = 100 \, \mu \mathrm{m} \) and \( D = 100 \, \mathrm{km} \). First, calculate \( \log 100 \) which is 2. Also, \( \log 100 \) for the distance is 2. So, the magnitude \( m = 2 - 2.48 + 2.76 \times 2 = 2 - 2.48 + 5.52 = 5.04 \).

Step 2: Calculate the Magnitude Most People Can Feel (Part b)

We need to calculate \( m \) for \( A = 1 \, \mathrm{mm} = 1000 \, \mu \mathrm{m} \) and \( D = 10 \, \mathrm{km} \). First, \( \log 1000 = 3 \) and \( \log 10 = 1 \). Substitute these into the formula: \( m = 3 - 2.48 + 2.76 \times 1 = 3 - 2.48 + 2.76 = 3.28 \).

Step 3: Calculate Amplitude for a Given Magnitude (Part c, i)

Given \( m = 7.2 \) and \( D = 10 \, \mathrm{km} \). Substitute into the formula: \( 7.2 = \log A - 2.48 + 2.76 \times 1 \). \( 7.2 = \log A - 2.48 + 2.76 \). Simplify to find \( \log A = 7.2 + 2.48 - 2.76 = 6.92 \). Thus, \( A = 10^{6.92} \approx 8.32 \times 10^6 \, \mu \mathrm{m} \).

Step 4: Calculate Amplitude for a Different Distance (Part c, ii)

Given \( m = 7.2 \) and \( D = 100 \, \mathrm{km} \). Substitute into the formula: \( 7.2 = \log A - 2.48 + 2.76 \times 2 \). Simplify: \( 7.2 = \log A - 2.48 + 5.52 \). Solving gives \( \log A = 7.2 + 2.48 - 5.52 = 4.16 \). So, \( A = 10^{4.16} \approx 1.45 \times 10^4 \, \mu \mathrm{m} \).

Step 5: Find the Amplitude Increase Factor (Part d)

If the magnitude increases by 1 (\( m+1 \)), setting this into the formula implies \( \log A_2 = \log A_1 + 1 \). This transformation corresponds to \( A_2 = 10 \cdot A_1 \). Thus, the factor increase in amplitude is 10 times.

Key Concepts

Earthquake Measurement

Earthquake measurement is crucial for understanding the strength and impact of earthquakes. Scientists use various methods to quantify these natural phenomena, and one widely adopted approach is the Richter magnitude scale.
This scale helps us express the magnitude of an earthquake in a manner that is understandable and comparable. It is calculated based on two significant factors:

• The amplitude of seismic waves, which indicates the physical shaking detected by a seismometer.
• The distance between the seismometer and the epicenter of the earthquake.

By evaluating these factors, the Richter scale produces a numerical value, allowing experts to gauge the size of earthquakes.
Monitoring allows geologists, engineers, and emergency services to respond adequately by anticipating potential risks and damages.

Logarithms in Mathematics

Logarithms play a fundamental role in the mathematical representation of earthquake magnitudes. They provide a way to manage the vast range of seismic data efficiently. Logarithms, especially common (base 10) logarithms, are a tool that converts multiplication into addition, which simplifies the analysis of exponential growth, such as seismic wave magnitudes.
In the context of earthquake measurement, the formula incorporates logarithms to account for the exponential nature of wave amplitude changes over different distances. For instance, the amplitude of ground shaking recorded by a seismometer is expressed in logarithmic form to allow for a linear interpretation, making analysis and comparison more intuitive.
This conversion helps convert the raw amplitude data to a comprehensible scale that reflects physical phenomena, ensuring scientists and researchers can communicate effectively with scalability and precision.

Seismometer Analysis

Seismometer analysis involves using sophisticated devices to detect, record, and measure seismic waves from earthquakes. These instruments are extremely sensitive and can pick up vibrations that are imperceivable to humans.
Seismometers capture seismic wave amplitude, which is crucial for Richter scale calculations. The amplitude represents the intensity of the shaking and serves as the primary input for the magnitude formula. For instance:

• A larger amplitude reading typically indicates a more powerful earthquake.
• The precision of the equipment ensures minute changes in wave amplitude are accurately documented.

The position of a seismometer relative to the earthquake epicenter is also pivotal as it influences the recorded amplitude. Thus, understanding both the limitations and capabilities of seismometers is vital for accurate earthquake analysis and magnitudes that reflect real-world conditions.