The magnitude of an earthquake, [tex]$R$[/tex], can be measured by the equation [tex]$R=\log \left(\frac{A}{T}\right)+D$[/tex], where [tex]$A$[/tex] is the amplitude in micrometers, [tex]$T$[/tex] is measured in seconds, and [tex]$D$[/tex] accounts for the weakening of the earthquake due to the distance from the epicenter.
If an earthquake occurred for 4 seconds and [tex]$D=2$[/tex], which graph would model the correct amount on the Richter scale?
Answer (1)
Substitute given values into the formula: R = lo g ( 4 A ) + 2 .
Simplify the equation using logarithm properties: R = lo g ( A ) − lo g ( 4 ) + 2 .
Approximate the constant: R = lo g ( A ) + 1.398 .
The graph is a logarithmic curve that increases as A increases: R = lo g ( A ) + 1.398 .
Explanation
Understanding the Problem We are given the formula for the magnitude of an earthquake: R = lo g ( T A ) + D , where A is the amplitude, T is the time in seconds, and D is a constant that depends on the distance from the epicenter. We are given that T = 4 seconds and D = 2 . We want to find the equation that models the Richter scale magnitude R as a function of the amplitude A .
Substituting the Values Substitute the given values of T and D into the formula: R = lo g ( 4 A ) + 2
Simplifying the Equation Using the properties of logarithms, we can rewrite the equation as: R = lo g ( A ) − lo g ( 4 ) + 2
Approximating the Constant Since lo g ( 4 ) is a constant, we can approximate its value: lo g ( 4 ) ≈ 0.602 . Therefore, the equation becomes: R = lo g ( A ) − 0.602 + 2 R = lo g ( A ) + 1.398
Analyzing the Function The equation R = lo g ( A ) + 1.398 represents a logarithmic function. The graph of this function will have the general shape of a logarithmic curve. As the amplitude A increases, the magnitude R also increases, but at a decreasing rate. The vertical shift of the graph is 1.398 . The domain of the function is 0"> A > 0 since the amplitude must be positive.
Conclusion Therefore, the graph that models the Richter scale magnitude R as a function of the amplitude A will be a logarithmic curve that increases as A increases.
Examples
Earthquake magnitude is a real-world application of logarithmic scales. For example, an earthquake with a magnitude of 6 is ten times stronger than an earthquake with a magnitude of 5. This concept is used by seismologists to measure and compare the strength of different earthquakes. Understanding logarithmic scales helps us comprehend the vast differences in energy released by earthquakes of varying magnitudes.
