Problem 62 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

On the Richter scale, the magnitude \(R\) of an earthquake is given by the formula $$ R=\log \frac{I}{I_{0}} $$ where \(I\) is the intensity of the earthquake being measured and \(I_{0}\) is the standard reference intensity. a. Express the intensity \(I\) of an earthquake of magnitude \(R=5\) in terms of the standard intensity \(I_{0}\). b. Express the intensity \(I\) of an earthquake of magnitude \(R=8\) in terms of the standard intensity \(I_{0}\). How many times greater is the intensity of an earthquake of magnitude 8 than one of magnitude \(5 ?\) c. In modern times, the greatest loss of life attributable to an earthquake occurred in eastern China in 1976 . Known as the Tangshan earthquake, it registered \(8.2\) on the Richter scale. How does the intensity of this earthquake compare with the intensity of an earthquake of magnitude \(R=5 ?\)

Step-by-Step Solution

Answer

The intensity of an earthquake with magnitude \(R=5\) is \(I = 10^5I_0\) and for an earthquake with magnitude \(R=8\), the intensity is \(I = 10^8I_0\). The intensity of an earthquake of magnitude 8 is 1000 times greater than that of an earthquake of magnitude 5. The intensity of the Tangshan earthquake, with a magnitude of 8.2, is around 1585 times greater than the intensity of an earthquake with a magnitude of 5.

Step 1: Write down given information and formula

We are given the magnitude of an earthquake \(R = 5\). The formula for the Richter scale magnitude is: $$R = \log\frac{I}{I_0}$$

Step 2: Solve for Intensity I

We want to find the intensity \(I\) for the given magnitude \(R=5\). To do this, we will isolate \(I\) from the given formula: $$5 = \log\frac{I}{I_0}$$ To get I, we can express this equation as an exponential by raising both sides of the equation to the base 10: $$10^5 = \frac{I}{I_0}$$ Now, multiply both sides by \(I_0\) to isolate \(I\): $$I = 10^5I_0$$ #a. Calculate intensity I for R=8#

Step 1: Write down given information and formula

We are given the magnitude of an earthquake \(R = 8\). The formula for the Richter scale magnitude is: $$R = \log\frac{I}{I_0}$$

Step 2: Solve for Intensity I

We want to find the intensity \(I\) for the given magnitude \(R=8\). As before, we will isolate \(I\) from the given formula: $$8 = \log\frac{I}{I_0}$$ To get I, we can express this equation as an exponential by raising both sides of the equation to the base 10: $$10^8 = \frac{I}{I_0}$$ Now, multiply both sides by \(I_0\) to isolate \(I\): $$I = 10^8I_0$$ #b. Compare the intensities of earthquakes with magnitudes R=5 and R=8#

Step 1: Calculate the ratio of earthquake intensities

We found the intensities of earthquakes for magnitudes \(R=5\) and \(R=8\). We will now calculate the ratio \(\frac{I_{8}}{I_{5}}\): $$\frac{10^8I_0}{10^5I_0}$$

Step 2: Simplify the ratio expression

Simplify the expression by canceling out \(I_0\) from the numerator and denominator, and then divide the powers of 10: $$\frac{10^8}{10^5} = 10^3$$ This means that the intensity of an earthquake of magnitude 8 is 1000 times greater than that of an earthquake of magnitude 5. #c. Compare the intensities of the Tangshan earthquake (R=8.2) with an earthquake of (R=5)#

Step 1: Calculate the intensity of the Tangshan earthquake

We are given the magnitude of the Tangshan earthquake \(R = 8.2\). We will find the intensity \(I\) using the formula: $$8.2 = \log\frac{I}{I_0}$$ To get I, we can express this equation as an exponential by raising both sides of the equation to the base 10: $$10^{8.2} = \frac{I}{I_0}$$ Now, multiply both sides by \(I_0\) to isolate \(I\): $$I = 10^{8.2}I_0$$

Step 2: Calculate the ratio of earthquake intensities

We found the intensities of the Tangshan earthquake and an earthquake with magnitude \(R=5\). We will now calculate the ratio \(\frac{I_{8.2}}{I_{5}}\): $$\frac{10^{8.2}I_0}{10^5I_0}$$

Step 3: Simplify the ratio expression

Simplify the expression by canceling out \(I_0\) from the numerator and denominator, and then divide the powers of 10: $$\frac{10^{8.2}}{10^5} = 10^{3.2}$$ This means that the intensity of the Tangshan earthquake with magnitude 8.2 is around 1585 times greater than the intensity of an earthquake with magnitude 5.

Key Concepts

Logarithmic Equations

Understanding logarithmic equations is essential when studying the Richter scale, which is used to measure the magnitude of earthquakes. A logarithmic equation has the form \( y = \log_b(x) \) where \( b \) is the base, \( x \) is the argument ('input' value), and \( y \) is the result or 'output' value. To solve for \( x \) when you have the result, you must use exponential form, converting the log equation into an exponent: \( b^y = x \). This change from a logarithmic to an exponential equation is what allows us to solve for earthquake intensity \( I \) on the Richter scale, as shown in the original exercise.

Intensity of Earthquakes

The intensity of an earthquake refers to the energy released during the seismic event and is a key factor in determining its magnitude. It's not directly measurable but is derived from the amplitude of the waves recorded by seismographs. The formula \( R = \log(\frac{I}{I_0}) \) links the magnitude \( R \) to the intensity \( I \) with respect to a standard intensity \( I_0 \). An important aspect to grasp is that this intensity isn't linear - small increases in \( R \) correspond to exponentially larger differences in \( I \) due to the logarithmic nature of the scale. This exponential relationship means that an earthquake's destructive potential can increase dramatically with a relatively small increase in magnitude.

Exponential Functions

Exponential functions play a pivotal role in translating the magnitude of an earthquake into its relative intensity. In mathematical terms, an exponential function is of the form \( y = b^x \) where \( b \) is the base and \( x \) is the exponent. These functions grow (or decay) at rates proportional to their current value, which is why they describe phenomena like interest growth, population growth, and, in our case, the increase in earthquake intensity. When we solve a logarithmic equation for intensity, we express the base-10 logarithm as a power of 10, which effectively gives us an exponential expression indicating the actual intensity.

Magnitude Comparison

Comparing the magnitudes of different earthquakes can give us insight into their relative power. The Richter scale is logarithmic, meaning each whole number step represents a tenfold increase in measurement of the amplitude of seismic waves, which corresponds to roughly 31.6 times more energy release. For example, a magnitude 8 earthquake isn't just three steps more than a magnitude 5, but in terms of intensity, it's 1,000 times (or \( 10^3 \) times) stronger, as the step-by-step solution illustrated. This is why magnitude comparison is not intuitive; a magnitude 8.2 earthquake, like the Tangshan earthquake, is much more devastating than a magnitude 5 earthquake - approximately 1,585 times greater in intensity.