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Chapter 3: Problem 9

How many more times intense is an earthquake with Richter magnitude 7 than an earthquake with Richter magnitude \(5 ?\)

Short Answer

Expert verified
The earthquake with Richter magnitude 7 is 100 times more intense than the earthquake with Richter magnitude 5.

Step by step solution

01

Understand the Richter scale formula

The Richter scale formula is given by: \(M_L = log_{10}(A / A_0)\) Where: - \(M_L\) is the Richter magnitude, - \(A\) is the amplitude of the seismic wave, and - \(A_0\) is a reference amplitude (generally the smallest amplitude that can be measured with instruments). The equation can be rearranged to make A the subject: \(A = A_0 * 10^{M_L}\) We will use this formula to calculate the ratio of the amplitudes of the two earthquakes.
02

Calculate the amplitudes of the two earthquakes

Let's calculate the amplitudes of the earthquakes with Richter magnitudes 7 and 5. For the earthquake with Richter magnitude 7: \(A_1 = A_0 * 10^7\) For the earthquake with Richter magnitude 5: \(A_2 = A_0 * 10^5\) Notice that both of these equations have \(A_0\) as a common factor.
03

Calculate the ratio of the amplitudes

To find how many times more intense the earthquake with magnitude 7 is compared to the one with magnitude 5, we need to find the ratio between their amplitudes: \(\frac{A_1}{A_2} = \frac{A_0 * 10^7}{A_0 * 10^5}\) Simplify the equation by cancelling out the \(A_0\) terms: \(\frac{A_1}{A_2} = \frac{10^7}{10^5}\)
04

Find the intensity ratio

Now we can use the properties of exponents to simplify the equation further: \(\frac{10^7}{10^5} = 10^{(7-5)} = 10^2 = 100\) Therefore, the earthquake with Richter magnitude 7 is 100 times more intense than the earthquake with Richter magnitude 5.

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