The 1985 Mexico City earthquake measured a magnitude 8.0 on the Richter scale. The Richter scale uses the function [tex]$M(I)=\log \left(\frac{I}{I_0}\right)$[/tex]. Which equation relates the intensity of the 1985 earthquake, [tex]$I$[/tex], with the intensity of the threshold quake, [tex]$I_0$[/tex] ?
A. [tex]$I=8^{10}\left(I_0\right)$[/tex]
B. [tex]$I_0=8^{10}(I)$[/tex]
C. [tex]$I=10^8\left(I_0\right)$[/tex]
D. [tex]$I_0=10^8(I)$[/tex]
Answer (1)
Substitute the magnitude into the Richter scale equation.
Rewrite the equation in exponential form.
Solve for I in terms of I 0 .
The equation relating the intensities is I = 1 0 8 ( I 0 ) .
Explanation
Understanding the Problem We are given that the magnitude of the 1985 Mexico City earthquake was 8.0 on the Richter scale. The Richter scale is defined by the function M ( I ) = lo g ( I 0 I ) , where I is the intensity of the earthquake and I 0 is the intensity of a threshold quake. We need to find the equation that relates I and I 0 .
Substituting the Magnitude Substitute the magnitude M ( I ) = 8.0 into the Richter scale equation: 8.0 = lo g ( I 0 I ) .
Rewriting in Exponential Form Since the logarithm is base 10, rewrite the equation in exponential form: 1 0 8 = I 0 I .
Solving for I Solve for I in terms of I 0 by multiplying both sides of the equation by I 0 : I = 1 0 8 I 0 .
Final Answer The equation that relates the intensity of the 1985 earthquake, I , with the intensity of the threshold quake, I 0 , is I = 1 0 8 I 0 .
Examples
Understanding the Richter scale helps us relate the intensity of an earthquake to a reference intensity. For example, if a seismograph measures an earthquake with a magnitude of 6.0, we can determine how much more intense it is compared to a threshold quake. This knowledge is crucial for assessing potential damage and preparing for future seismic events. By comparing different earthquakes, we can better understand the energy released and the potential impact on communities.
