Are the answers correct?
The Richter scale is used to measure the intensity of an earthquake. If E, measured in joules, is the energy released during an earthquake, then the magnitude is given by,
M = 2/3 log(E/E_0)
Where E_0 = 10^4.4 joules
1. If 8 x 10^14 joules of energy are released during an earthquake, what is the magnitude of the earthquake?
answer: M = 2/3 log_10((8×10^14)/(1×10^4.4)) =~7 - Magnitude on the Richter scale
2. How much energy will be released in an earthquake with a magnitude of 5.9?
Answer: 5.9 = 2/3 log_10(P/(1×10^4.4))
P = 1.77828×10^13
- Joules - the amount of energy released in a 5.9 Magnitude earthquake.
To verify the correctness of the answers, we can go through the calculations step by step.
1. The magnitude of the earthquake is given by M = 2/3 log(E/E_0), where E is the energy released during the earthquake and E_0 is a constant equal to 10^4.4 joules.
For the first question, we are given that E = 8 x 10^14 joules. We need to substitute this value into the equation to calculate the magnitude.
M = 2/3 log((8 x 10^14)/(10^4.4))
First, we simplify the ratio inside the log:
M = 2/3 log(8 x 10^14 / 10^4.4)
M = 2/3 log(8 x 10^(14-4.4))
M = 2/3 log(8 x 10^9.6)
Next, we can use the property of logarithms to simplify further:
M = 2/3 [log(8) + log(10^9.6)]
M = 2/3 [log(8) + 9.6]
Now, we can calculate the logarithm and simplify:
M = 2/3 [0.903 + 9.6]
M = 2/3 x 10.503
M ≈ 7
Therefore, the magnitude of the earthquake with 8 x 10^14 joules of energy is approximately 7 on the Richter scale. This matches the answer provided.
2. For the second question, we are given a magnitude of 5.9, and we need to calculate the amount of energy released (P).
The equation is given by 5.9 = 2/3 log(P/10^4.4). We can rearrange the equation to isolate P:
5.9 = 2/3 log(P/10^4.4)
5.9 x 3/2 = log(P/10^4.4)
Logarithmic equations can be solved using the exponential function. By applying an exponential with base 10 on both sides of the equation, we can solve for P:
10^(5.9 x 3/2) = P/10^4.4
First, simplify the exponent and solve:
10^(8.85) = P/10^4.4
7.94328235 x 10^8 = P/10^4.4
Now, multiply both sides by 10^4.4 to isolate P:
P = 7.94328235 x 10^8 x 10^4.4
P ≈ 1.77828 x 10^13
Therefore, the amount of energy released in an earthquake with a magnitude of 5.9 is approximately 1.77828 x 10^13 joules. This matches the answer provided.
Based on the step-by-step calculation, we can conclude that the answers provided are correct.