Problem 49 - Calculus and its applications

Question

The intensity of an earthquake is given by \(I=I_{0} 10^{R},\) where \(R\) is the magnitude on the Richter scale and \(I_{0}\) is the minimum intensity, at which \(R=0,\) used for comparison. a) Find \(I\), in terms of \(I_{0}\), for an earthquake of magnitude 7 on the Richter scale. b) Find \(I,\) in terms of \(I_{0},\) for an earthquake of magnitude 8 on the Richter scale. c) Compare your answers to parts (a) and (b). d) Find the rate of change \(d I / d R\). e) Interpret the meaning of \(d I / d R\).

Step-by-Step Solution

Answer

(a) \(I = I_0 10^7\); (b) \(I = I_0 10^8\); (c) Intensity for magnitude 8 is ten times that for magnitude 7; (d) \(dI/dR = I_0 \ln(10) 10^R\); (e) Describes the exponential growth of intensity with magnitude.

Step 1: Understanding the Formula

The intensity of an earthquake is modeled by the formula \(I = I_{0} 10^{R}\). Here, \(I\) is the intensity, \(I_{0}\) is the reference intensity when the magnitude \(R = 0\), and \(R\) represents the Richter scale magnitude.

Step 2: Calculate Intensity for Magnitude 7

For a magnitude \(R = 7\), substitute into the formula: \(I = I_0 10^{7}\). Thus, the intensity is \(I = I_0 10^{7}\).

Step 3: Calculate Intensity for Magnitude 8

For a magnitude \(R = 8\), substitute into the formula: \(I = I_0 10^{8}\). So, the intensity is \(I = I_0 10^{8}\).

Step 4: Compare Intensities for Magnitude 7 and 8

The intensity for magnitude 8, \(I = I_0 10^{8}\), is ten times greater than the intensity for magnitude 7, \(I = I_0 10^{7}\). This demonstrates the logarithmic nature of the Richter scale, where each whole number increase means a tenfold increase in measured amplitude.

Step 5: Find the Rate of Change \(dI/dR\)

The rate of change of intensity with respect to magnitude \(R\) is found by differentiating the formula \(I = I_0 10^{R}\) with respect to \(R\).Use: \(\frac{d}{dR}(I) = I_0 \frac{d}{dR}(10^R) = I_0 \ln(10) 10^R\), where \(\ln(10)\) is the natural logarithm of 10.

Step 6: Interpret the Rate of Change \(dI/dR\)

The expression \(\frac{dI}{dR} = I_0 \ln(10) 10^R\) indicates how rapidly the earthquake intensity changes with respect to changes in the Richter scale magnitude. A small increase in \(R\) results in a rapid increase in intensity due to the exponential dependency.

Key Concepts

Earthquake Intensity

The intensity of an earthquake is a measure of how much energy is released at the source of the earthquake. It helps to communicate how powerful an earthquake is in terms of its ability to cause damage. The formula used to calculate the intensity is given by:\[ I = I_{0} 10^{R} \]where:

• \(I\) is the intensity of the earthquake.
• \(I_{0}\) is the base intensity, representing the minimum level of intensity when \(R = 0\).
• \(R\) is the magnitude of the earthquake on the Richter scale.

In simple terms, a higher intensity means a more powerful earthquake. The formula shows that intensity increases exponentially as the Richter scale magnitude increases. This means even small increases in magnitude can lead to significant rises in intensity.

Logarithmic Scale

The Richter scale is a logarithmic scale used to measure earthquake magnitudes. Unlike linear scales, a logarithmic scale represents values using logarithms, which allows for a wide range of values to be condensed into a manageable scale. With each whole number increase on the Richter scale, the earthquake's intensity actually increases tenfold. For example:

• An earthquake with a magnitude of 7 is ten times more intense than one with a magnitude of 6.
• A magnitude 8 is ten times more intense than a magnitude 7.

This logarithmic nature is why earthquakes are classified this way—it provides a simple way to represent very large differences in energy release between earthquakes.

Differentiation

Differentiation is a mathematical technique used to determine how a function changes as its input changes. In the context of the Richter scale and earthquake intensity, differentiation allows us to find the rate at which intensity changes with respect to changes in the magnitude \(R\).When you differentiate the intensity formula \(I = I_{0} 10^{R}\) with respect to \(R\), you use the chain rule and the fact that the derivative of \(10^{R}\) with respect to \(R\) is \(\ln(10) \cdot 10^{R}\). Thus, the rate of change is given by:\[ \frac{dI}{dR} = I_0 \ln(10) \cdot 10^{R} \]This derivative indicates how the intensity increases exponentially with an increase in \(R\). Small changes in \(R\) can lead to large changes in \(I\) due to this exponential relationship.

Rate of Change

The concept of rate of change in this exercise pertains to how quickly the earthquake intensity changes as the Richter scale magnitude changes. It provides insight into the sensitivity of the intensity to changes in magnitude.The calculated rate of change is:\[ \frac{dI}{dR} = I_0 \ln(10) \cdot 10^{R} \]This result demonstrates that:

• For every small increase in \(R\), there is a large increase in intensity \(I\).
• The factor \(\ln(10)\) signifies how the base-10 logarithmic nature affects the rate.

In essence, this relationship hints at the explosive growth in intensity with even slightly larger earthquakes. This is important for assessing the potential impacts of earthquakes, emphasizing why small increments in the Richter scale magnitude can have dramatic effects.