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Chapter 3: Problem 10

How many more times intense is an earthquake with Richter magnitude 6 than an earthquake with Richter magnitude \(3 ?\)

Short Answer

Expert verified
An earthquake with a Richter magnitude of 6 is \(10^3\) or 1000 times more intense than an earthquake with a Richter magnitude of 3.

Step by step solution

01

Understand the Richter magnitude scale formula

The Richter magnitude scale formula is: \(M = log_{10}(A/A_0)\) Where: - M is the earthquake's magnitude - A is the amplitude of the earthquake's ground motion (in micrometers) - \(A_0\) is the amplitude of the reference earthquake (typically with a magnitude of 0)
02

Calculate the amplitude of the two earthquakes

We can rearrange the Richter magnitude scale formula to find the amplitude of each earthquake: \(A = A_0 \times 10^M\) We're comparing the two earthquakes to each other, so the reference amplitude (\(A_0\)) will cancel out. Thus, we have: \(A_1 = 10^{M_1}\) \(A_2 = 10^{M_2}\) Where: - \(A_1\) and \(A_2\) are the amplitudes of the two earthquakes - \(M_1 = 3\) and \(M_2 = 6\) are their magnitudes
03

Calculate the ratio of the amplitudes

To determine how many more times intense the magnitude 6 earthquake is compared to the magnitude 3 earthquake, we take the ratio of their amplitudes (\(A_2\) to \(A_1\)) : \(\frac{A_2}{A_1} = \frac{10^{M_2}}{10^{M_1}}\)
04

Plug in the magnitudes and calculate the ratio

Now we plug in the magnitudes of the two earthquakes: \(\frac{A_2}{A_1} = \frac{10^{6}}{10^{3}}\) Using the properties of exponents, we can simplify this expression: \(\frac{10^{6}}{10^{3}} = 10^{6-3} = 10^3\)
05

Interpret the result

The result, \(10^3\), means that the intensity (amplitude) of an earthquake with a Richter magnitude of 6 is 1000 times more intense than an earthquake with a Richter magnitude of 3.

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