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Question 1161102: An aftershock measuring 5.5 on the Richter scale occurred south of Christchurch, New Zealand in
June 2011. The magnitude,
M
of an earthquake that is
T
times more intense than an earthquake
measuring 5.5 on the Richter scale can be modeled by the following function:
M = log(T) + 5.5
every one unit of increase on the Richter scale, the intensity of an earthquake increases 10 fold.
a) Graph the function, [show window] and determine how much more intense an earthquake
measuring 7.7 that occurred on Queen Charlotte Islands in October 2012 was than the
aftershock in New Zealand.
b) Determine the value of the function at T=3 explain the answer in the context of this question
c) How many times as intense would an earthquake measuring 8.5 on the richer scale be then one measuring 5.5? Express your answer as a power with base 10.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! M = magnitude.
T = intensity.
M = log(T) + 5.5 = number of times more intense than an earthquake measuring 5.5 on the richter scale.
for every unit increase on the richter scale, the intensity of an earthquake increases 10 times.
the log function is defined as:
log(x) = y if and only if 10^y = x
therefore:
log(1) = y if and only if 10^y = 1, making y = 0 because 10^0 = 1
log(10) = y if and only if 10^y = 10, making y = 1 because 10^1 = 10
log(100) = y if and only if 10^y = 100, making y = 2 because 10^2 = 100.
and so on.....
you can see that, every time x is multiplied by 10, y is increased by 1.
back to your problem:
you are asked to provide solutions to the following:
a) Graph the function, [show window] and determine how much more intense an earthquake
measuring 7.7 that occurred on Queen Charlotte Islands in October 2012 was than the
aftershock in New Zealand.
here's the graph:
y replaces M and x replaces T in the graph.
y is the magnitude, same as M.
x is the intensity, same as T.
you can see from the graph, that, when y = 5.5, x = 1, and when y = 7.7, x = 158.489.
this tells you that the magnitude of an earthquake of 7.7 on the richter scale is 158.489 / 1 = 158.489 times as powerful as the magnitude of an earthquake of 1.5 on the richter scale.
the calculations are as follows, using M and T for the variables.
M and y represent the same thing, which is the magnitude.
T and x represent the same thihng, which is the intensity.
when y = 5.5, M = log(T) + 5.5 becomes 5.5 = log(T) + 5.5
subtract 5.5 from both sides of the equation to get 0 = log(T)
by properties of logs, 0 = log(T) if and only if 10^0 = T
solve for T to get T = 1.
the formula becomes 5.5 = log(1) + 5.5
when y = 7.7, M = log(T) + 5.5 becomes 7.7 = log(T) + 5.5
subtract 5.5 from both sides of the equation to get 2.2 = log(T)
by properties of logs, 2.2 = log(T) if and only if 10^2.2 = T
solve for T to get T = 10^2.2 = 158.4893192.
158.4893192 / 1 = 158.4893192.
T = 158.4893192 is 158.4893192 times as powerful as T = 1.
(158.4893192 - 1) / 1 = 157.4893192.
T = 158.4893192 is 157.4893192 times more powerful than T = 1
b) Determine the value of the function at T=3 explain the answer in the context of this question.
when T = 3, this tells you that the magnitude is 3 times as powerful at an earthquake with a magnitude of 5.5, because an earthquake with a magnitude of 5.5 (M = 5.5) has T = 1.
c) How many times as intense would an earthquake measuring 8.5 on the richer scale be then one measuring 5.5? Express your answer as a power with base 10.
M = log(T) + 5.5
when M = 5.5, the equation becomes 5.5 = log(T) + 5.5
subtract 5.5 from both sides of the equation to get 0 = log(T)
log(T) = 0 if and only if 10^0 = T
solve for T to get T = 1
this is because 10^0 = 1
when M = 8.5, the equation becomes 8.5 = log(T) + 5.5
subtract 5.5 from both sides of the equation to get 3 = log(T)
log(T) = 3 if and only if 10^3 = T
solve for T to get T = 1000
this is because 10^3 = 1000
the richter scale is a logarithmic scale.
the basic property of logs is that y = log(x) if and only if 10^y = x.
here's some good references on logs and exponents you might find helpful.
https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/
the lessons on exponents and logarithms are in tutorials 42 through 47.
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