|APPLICATIONS OF EXPONENTIAL|
EARTHQUAKE WORD PROBLEMS:
As with any word problem, the trick is convert a narrative statement or question to a mathematical statement.
Before we start, let's talk about earthquakes and how we measure their intensity.
In 1935 Charles Richter defined the magnitude of an earthquake to be
The magnitude of a standard earthquake is
Richter studied many earthquakes that occurred between 1900 and 1950. The largest had magnitude of 8.9 on the Richter scale, and the smallest had magnitude 0. This corresponds to a ratio of intensities of 800,000,000, so the Richter scale provides more manageable numbers to work with.
Each number increase on the Richter scale indicates an intensity ten times stronger. For example, an earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. An earthquake of magnitude 7 is times strong than an earthquake of magnitude 5. An earthquake of magnitude 8 is times stronger than an earthquake of magnitude 5.
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where I1 is the intensity of the larger earthquake and I2 is the intensity of the smaller earthquake.
We are trying to determine the ratio of the larger magnitude M1 to the smaller magnitude I2 or M1-M2. The reason we are subtracting the magnitudes instead of dividing them is the question asked how much larger, not how many times larger.
Solve for I1 by multiplying both sides of the equation by I2.
We can write M1-M2 as and we can write
The larger earthquake had a magnitude 1.4 more on the Richter scale than the smaller earthquake.
Let's check our answer:
Convert both of these equations to exponential equations.
What you are looking for is the ratio of the intensities: So our task is to isolate this ratio from the above given information using the rules of logarithms.
Convert the logarithmic equation to an exponential equation.
The stronger earthquake was 40 times as intense as the weaker earthquake.
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