The Richter scale measures earthquakes using the formula M =
2/3log(E/10^(4.4)) where M is the magnitude of the earthquake that releases E joules of energy.
(a) Find the magnitude of an earthquake that releases 1013.4
joules of energy.
(b) Find how much energy is released by an earthquake that has a magnitude of 7.2
(a) just find M(1013.4)
(b) find E such that 2/3 log(E/10^(4.4)) = 7.2
To find the magnitude of an earthquake that releases a certain amount of energy, we can use the given formula:
M = (2/3) * log(E / 10^(4.4))
where M is the magnitude of the earthquake and E is the amount of energy released in joules.
(a) Find the magnitude of an earthquake that releases 1013.4 joules of energy:
Substitute E = 1013.4 into the formula:
M = (2/3) * log(1013.4 / 10^(4.4))
Simplify the expression inside the logarithm:
M = (2/3) * log(101.34)
Evaluate the logarithm using the base 10 logarithm:
M = (2/3) * log10(101.34)
Using a calculator, find the value of log10(101.34). Let's assume it equals 2.005.
M = (2/3) * 2.005
Now, calculate the final magnitude:
M ≈ 1.3367
Therefore, the magnitude of an earthquake that releases 1013.4 joules of energy is approximately 1.3367.
(b) Find how much energy is released by an earthquake that has a magnitude of 7.2:
We need to rearrange the formula to solve for E:
M = (2/3) * log(E / 10^(4.4))
Multiply both sides by 3/2:
(3/2) * M = log(E / 10^(4.4))
Rewrite the equation using exponential form:
10^((3/2) * M) = E / 10^(4.4)
Multiply both sides by 10^(4.4):
10^((3/2) * M + 4.4) = E
Substitute M = 7.2 into the equation:
10^((3/2) * 7.2 + 4.4) = E
Evaluate the expression using a calculator:
10^(10.8 + 4.4) = E
10^15.2 = E
E ≈ 1629779.664
Therefore, an earthquake with a magnitude of 7.2 releases approximately 1,629,779.664 joules of energy.