Maya Vaughan
1/25/20
Pre Calc Pow #7
Problem Statement:
The formula for the Richter scale number of an earthquake is R=log₁₀ a. A is the amplitude of the earthquake, and R is the Richter scale number of the earthquake.
1a. How many times as much ground motion does a 8.3 quake have compared to a 4.0 quake?
1b. What Richter scale measurement would represent a quake that has twice the ground motion of one that measures 4.0 on the Richter scale?
1c. What Richter scale number would represent a quake that has half the ground motion of one that measures 8.3 on the Richter scale?
2a. The 1989 Loma Prieta earthquake was measured at 7.1 on the Richter scale. In numerical terms, how did the amount of of its ground motion compare to that of the 1906 earthquake?
2b. Write a general formula that allows you to find the difference in amplitude between two earthquakes, given their Richter scale numbers.
Process:
To start this POW, we learned about logarithms in class, and we worked through question 1a as a class. We used the Richter scale numbers that we were given for both earthquakes and plugged them into the equation in the problem statement. When we did this, we were able to find the amplitude of both earthquakes. The amplitude of the 4.0 earthquake was 10,000 and the amplitude of the 8.3 earthquake was 199,526,231.5. Then I divided the amplitude of the 8.3 earthquake by the amplitude of the 4.0 earthquake to find how many times more ground motion it has. The work for that is shown below:
Then I started on question 1b. To answer this question, I needed to know what the amplitude of a 4.0 earthquake was, and I figured that out in the last question. The amplitude of a 4.0 is 10,000 so double that would be 20,000. So, I needed to figure out what an earthquake with an amplitude of 20,000 would register on the Richter scale. So, I plugged 20,000 into the given equation like this: R=log₁₀ (20,000), and I solved it on the calculator. I got an answer of 4.3. So, an earthquake with double amplitude of a 4.0 would register as 4.3 on the Richter scale.
Next, I did question 1c. This question was a lot like 1b. I knew that an 8.3 earthquake has an amplitude of 199,526,231.5, so I divided it by 2 to find the amplitude of an earthquake half as strong as an 8.3. The answer that I got was 99,763,115.75. So, I plugged this number into the given equation like this: R=log₁₀ (99,763,115.75), and I solved it with a calculator and got the answer 7.99897, which I rounded up to an 8.0. So, an earthquake with half the amplitude of an 8.3 is an 8.0.
Next, I answered question 2a. I already knew that the 1906 earthquake measured an 8.3, which has an amplitude of 199,526,231.5. Now, I needed to find the amplitude of the Loma Prieta earthquake which measured a 7.1. To do this, I plugged 7.1 into the given equation, and then solved. The answer that I got was 12,589,254.12. This means that the Loma Prieta earthquake had an amplitude of 12,589,254.12. Now I needed to find the difference between these two amplitudes. So I divided 199,526,231.5 by 12,589,254.12 and got 15.85. This means that the 1906 earthquake was 15.85 times stronger than the Loma Prieta earthquake.
Last, I had to solve question 2b. I worked on this question for a long time and I found equations for specific differences but i could not find a general equation.
Solution:
1a. The 8.3 earthquake has 19,952.62315 times as much ground motion as the 4.0 earthquake.
1b. A 4.3 earthquake would have twice as much ground motion as a 4.0 earthquake.
1c. An 8.0 earthquake would have half as much ground motion as an 8.3 earthquake.
2a. The ground motion in the 1906 earthquake was 15.85 times stronger than the ground motion in the Loma Prieta earthquake.
2b. No answer.
Reflection:
I really enjoyed this POW because it helped me to better understand logarithms and it also taught me how to use logarithms in both directions, and convert them from logarithmic form to exponential form, and back the other direction. I also thought that this POW was fun and interesting to solve because it incorporated a real world example that was interesting to me. I would like to go over this POW in class so that I can figure out the answer to question 2b, because I am curious how you can create a general formula for this. One habit of a mathematician that I used during this POW was reflecting and synthesizing because I had to take my numerical answers and figure out what their meaning was to solve the problems.
Extension Question:
Find formulas for the difference between two specific earthquakes. For example, write a formula for the difference in ground movement between the 1906 earthquake and the Loma Prieta earthquake.