1. How many times more intense is an 8.6 earthquake on the Richter scale, compared to a 7.2 earthquake?
Solve using exponents only.
2. What is the Richter scale measurement for an earthquake that is 100 times more intense than a 5.0
earthquake? Solve using logs only.
1. 8.6^2 / 7.2^2 = 10.76
2. log(100) + 5.0 = 8.0
To answer both of these questions, we can use the logarithmic relationship between earthquake magnitudes on the Richter scale. The Richter scale is logarithmic, which means that each increase of one magnitude represents a tenfold increase in amplitude (energy released by the earthquake).
1. To find how many times more intense an 8.6 earthquake is compared to a 7.2 earthquake, we need to calculate the ratio of their amplitudes.
Ratio = 10^(magnitude2 - magnitude1)
Therefore, the ratio of the intensities can be calculated as:
Ratio = 10^(8.6 - 7.2)
Using exponents, we have:
Ratio = 10^1.4
Calculating this, we get:
Ratio ≈ 25.1189
This means that an 8.6 earthquake is approximately 25.12 times more intense than a 7.2 earthquake.
2. To find the Richter scale measurement for an earthquake that is 100 times more intense than a 5.0 earthquake, we need to determine the magnitude using logarithms.
Using the formula for logarithms, we have:
magnitude2 = log(base 10)(intensity2 / intensity1)
In this case, the intensity2 is 100 times greater than intensity1, so we can substitute those values:
magnitude2 = log(base 10)(100 / 1)
Simplifying further, we have:
magnitude2 = log(base 10)(100)
Using logarithms to base 10, we find:
magnitude2 ≈ 2
Therefore, the Richter scale measurement for an earthquake that is 100 times more intense than a 5.0 earthquake is approximately 2.
1. To determine how many times more intense an 8.6 earthquake is compared to a 7.2 earthquake using exponents only, we can use the formula:
Intensity ratio = 10^(1.5 * (magnitude2 - magnitude1))
In this case, magnitude1 = 7.2 and magnitude2 = 8.6. Substituting these values into the formula:
Intensity ratio = 10^(1.5 * (8.6 - 7.2))
= 10^(1.5 * 1.4)
= 10^(2.1)
≈ 1258.925
Therefore, an 8.6 earthquake is approximately 1258.925 times more intense than a 7.2 earthquake.
2. To determine the Richter scale measurement for an earthquake that is 100 times more intense than a 5.0 earthquake using logs only, we can use the formula:
magnitude2 = magnitude1 + log10(intensity ratio)
In this case, magnitude1 = 5.0 and intensity ratio = 100. Substituting these values into the formula:
magnitude2 = 5.0 + log10(100)
= 5.0 + 2
= 7.0
Therefore, the Richter scale measurement for an earthquake that is 100 times more intense than a 5.0 earthquake is 7.0.