Problem 84 - Precalculus : Building Concepts and Connections

Question

Refer to the following. The magnitude of an earthquake is measured on the Richter scale using the formula $$R(I)=\log \left(\frac{I}{I_{0}}\right)$$ where I represents the actual intensity of the earthquake and \(I_{0}\) is a baseline intensity used for comparison. Richter Scale If the intensity of an earthquake is a million times the baseline intensity \(I_{0},\) what is its magnitude on the Richter scale?

Step-by-Step Solution

Answer

The earthquake's magnitude on the Richter scale is 6.

Step 1: Understand the provided formula and given value

The formula given is \(R(I)=\log \left(\frac{I}{I_{0}}\right)\) where \(R(I)\) represents the Richter scale magnitude, \(I\) represents the actual intensity of the earthquake, \(I_{0}\) is the baseline intensity. The problem tells us that \(I\) is a million times \(I_{0}\). So we can denote this as \(I= 1,000,000 \times I_{0}\).

Step 2: Substitute the values into the formula

Now we substitute \(I= 1,000,000 \times I_{0}\) into the Richter scale calculation formula: \(R(I)=\log \left(\frac{1,000,000 \times I_{0}}{I_{0}}\right)\).

Step 3: Simplify the formula

By simplifying, the \(I_0\) in the numerator and denominator cancel out, leaving us with \(R(I)=\log(1,000,000)\).

Step 4: Calculate the logarithm

Finally, we calculate the logarithm to base 10 of 1,000,000. The logarithm to base 10 of 1,000,000 is 6 because \(10^6 = 1,000,000\). So, \(R(I) = 6\).

Key Concepts

Understanding Earthquake Magnitude

Earthquake magnitude is a way to measure the size or energy release of an earthquake. It's a crucial aspect because it helps scientists and emergency services determine the potential impact and related hazards. The magnitude gives us a numerical value that tells us how much energy the earthquake has released. On the Richter scale, this value is represented through a number that typically ranges between 0 and 10, although it can technically go higher or lower depending on the size of the earthquake.

- A greater magnitude indicates a stronger earthquake, with more significant potential damage. - Each increment of 1 on the Richter scale represents a tenfold increase in amplitude and roughly 31.6 times more energy release.
The Richter scale, which was developed in 1935, is designed so that very small earthquakes have low magnitudes, while incredibly large ones have very high magnitudes. One important point here is the logarithmic nature of the scale, making even small changes in magnitude quite significant.

Demystifying Logarithmic Functions

Logarithmic functions are a key element in understanding the mathematics behind the Richter scale. These functions are the inverse of exponential functions, which means they help us find the power to which a number must be raised to get another number.

In simple terms, if we have a number like 100 and we want to find out what power we should raise 10 to get 100, we use a logarithm. Here, the base 10 logarithm of 100 is 2 because 10 squared (\(10^2\)) is 100.
- The Richter scale formula is a perfect example showcasing logarithmic functions: \(R(I)=\log \left(\frac{I}{I_{0}}\right)\). - Essentially, it shows how many times the earthquake's intensity \(I\) is stretched over the baseline intensity \(I_{0}\).
Logarithmic scales are advantageous in this context because they can effectively translate vast ranges of data into more manageable figures. The use of base 10 is especially significant when dealing with phenomena like earthquakes that can vary widely in energy output.

Exploring Intensity Measurement

Intensity measurement in the context of earthquakes refers to the quantification of the shaking's strength and the earthquake's energy. This measurement is crucial in the formulation and understanding of the Richter scale. The intensity, denoted as \(I\), represents how much shaking is observed at a particular location. This can vary widely depending on the earthquake's distance, depth, and geological conditions.

- The baseline intensity \(I_{0}\) is a reference point used to compare the actual intensity.- On the Richter scale, the intensity is often compared to predefined baseline, which standardizes the measurement.
In the original exercise, the intensity \(I\) was described as a million times the baseline intensity \(I_{0}\). When this relationship is plugged into the formula, it simplifies to calculating the logarithm of a million to the base 10. Understanding this relationship is vital for interpreting the significance of a reported earthquake's magnitude.
The accurate measurement and interpretation of intensity help ensure that seismic activities are assessed correctly, enabling appropriate responses to potential hazards.