The magnitude, m, of an earthquake is defined to be m = log startfraction i over s endfraction, where i is the intensity of the earthquake (measured by the amplitude of the seismograph wave) and s is the intensity of a “standard” earthquake, which is barely detectable. what is the magnitude of an earthquake that is 35 times more intense than a standard earthquake? use a calculator. round your answer to the nearest tenth. –1.5 –0.5 1.5 3.6
The Correct Answer and Explanation is:
To solve this problem, we need to use the given formula for the magnitude ( m ) of an earthquake:
[
m = \log \left( \frac{i}{s} \right)
]
Where:
- ( m ) is the magnitude of the earthquake.
- ( i ) is the intensity of the earthquake (measured by the amplitude of the seismograph wave).
- ( s ) is the intensity of a “standard” earthquake, which is barely detectable.
The problem states that the earthquake is 35 times more intense than the standard earthquake, so we can express this as:
[
i = 35s
]
Substitute this into the formula for ( m ):
[
m = \log \left( \frac{35s}{s} \right)
]
Simplifying the fraction:
[
m = \log (35)
]
Next, we need to calculate the logarithm of 35. Using a calculator:
[
\log (35) \approx 1.544
]
Rounding this value to the nearest tenth:
[
m \approx 1.5
]
Therefore, the magnitude of the earthquake is 1.5.
Explanation:
The formula ( m = \log \left( \frac{i}{s} \right) ) is based on a logarithmic scale. The logarithmic scale is used to represent large variations in the intensity of earthquakes in a more manageable form. In this case, we are comparing the intensity of the earthquake to that of a standard earthquake, which has a very small intensity value (( s )).
The problem tells us that the earthquake is 35 times more intense than the standard earthquake. This means that the ratio of ( i ) (the intensity of the earthquake) to ( s ) (the standard intensity) is 35. Taking the logarithm of this ratio gives us the magnitude of the earthquake. Since logarithms are a powerful way to deal with large differences in numbers, they allow us to express the magnitude of earthquakes in a compact and manageable way.
The final answer of 1.5 indicates that the earthquake is significantly stronger than the barely detectable standard earthquake.