Question 1112055: The Richter scale measures the energy released at the centre of an earthquake, (intensity), of earthquakes and is measured according to the equation:  , and if comparing two earthquakes,  where I is the intensity of the earthquake and S is the intensity of a "standard" earthquake. This is a logarithmic scale which runs from 1 to 9.5, where 7 is 10 times more intense than 6 and 100 times more intense than 5. In 1964 Alaska had an earthquake with a magnitude of 9.2 on the Richter scale. How many times more intense was the Alaska earthquake than the 1994 California earthquake which had a magnitude of 6.4 and caused between $15 billion to $30 billion in damage and killed 57 people?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula for richter scale is R = (I/S)
let the intensity of the alaska earthquake be represented by A.
let the intensity of the california earthquake be represented by C.
the formula for alaska becomes 9.5 = log(A/S).
the formula for california becomes 6.4 = log(C/S).
the definition of logs states that y = logb(x) if and only if b^y = x.
remember that log, by itself, means log10, which means log to the base of 10.
therefore y = log(x) is the same asy = log10(x) and the logarithm definition states that:
y = log10(x) if and only if 10^y = x.
using this definition for alaska gets us:
9.5 = log(A/S) if and only if 10^9.5 = A/S.
solve for A to get A = S * 10^9.
using this definition for california gets us:
6.4 = log(C/S) if and only if 10^6.4 = C/S.
solve for C to get C = S * 10^6.4
how many times the intensity of the alaska earthquake is compared to the intensity of the california earthqube becomes A/C.
A/C becomes (10^9.5 * S) / (10^6.4 * S)
the S in the numerator and the denominator cancel out and you are left with:
A/C = 10^9.5 / 10^6.4.
since b^x / b^y = b^(x-y), then you get:
10^9.5 / 10^6.4 = 10^(9.5-6.4) which results in A/C = 10^3.1.
solve for A/C to get A/C = 1258.925412.
knowing how the formulas work, you could have simply subtracted 6.4 from 9.5 and gotten 3.1 and then taken 10^3.1 directly to get the same answer.
remember, each increase of 1 on the richter scale is 10 times more powerful.
going from 6.4 to 7.4 gets you 7.4 is 10^1 times as powerful as 6.4.
going from 7.4 to 8.4 get you 8.4 is 10^1 times as powerful as 7.4 which is 10^1 times as powerful as 6.4.
going from 8.4 to 9.4 gets you 9.4 is 10^1 times as powerful as 8.4 which is 10^1 times as powerful as 7.4 which is 10^1 times as powerful as 6.4.
going from 9.4 to 9.5 gets you 9.5 is 10^.1 times as powerful as 9.4 which is 10^1 times as powerful as 8.4 which is 10^1 times as powerful as 7.4 which is 10^1 times as powerful as 6.4
what this says is that 9.5 is 10^.1 * 10^1 * 10^1 * 10^1 as powerful as 6.4.
if you remember your exponent arithmetic formulas, you know that b^x * b^y = b^(x+y), therefore 10^.1 * 10^1 * 10^1 * 10^1 is equal to 10^(.1 + 1 + 1 = 1) which is equal to 10^3.1 which is equal to 1258.925412.
the intensity of the alaska earthquake was 1258.925412 times the intensity of the california earthquake.
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