Answered

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The magnitude, [tex]\( M \)[/tex], of an earthquake is defined to be [tex]\( M=\log \frac{I}{S} \)[/tex], where [tex]\( I \)[/tex] is the intensity of the earthquake (measured by the amplitude of the seismograph wave) and [tex]\( S \)[/tex] is the intensity of a "standard" earthquake, which is barely detectable.

Which equation represents the magnitude of an earthquake that is 10 times more intense than a standard earthquake?

A. [tex]\( M=\log \frac{10 I}{S} \)[/tex]
B. [tex]\( M=\log (10 S) \)[/tex]
C. [tex]\( M=\log \frac{10 S}{S} \)[/tex]
D. [tex]\( M=\log \frac{10}{S} \)[/tex]

Sagot :

GinnyAnswer
First, let's understand the given formula for the magnitude of an earthquake:

[tex]\[ M = \log \frac{1}{S} \][/tex]

where:
- [tex]\( M \)[/tex] is the magnitude of the earthquake
- [tex]\( S \)[/tex] is the intensity of a standard, barely detectable earthquake

Now, we need to determine the magnitude of an earthquake that is 10 times more intense than a standard earthquake.

Given that the intensity of the earthquake ([tex]\( I \)[/tex]) is 10 times the intensity of a standard earthquake ([tex]\( S \)[/tex]), we can express this relationship as:

[tex]\[ I = 10 \times S \][/tex]

We need to find the appropriate equation for the magnitude of this earthquake.

Starting from the definition of magnitude [tex]\( M = \log \frac{1}{I} \)[/tex], let's substitute [tex]\( I = 10 \times S \)[/tex]:

[tex]\[ M = \log \frac{1}{10 \times S} \][/tex]

Now, let's write this equation in a simpler form:

[tex]\[ M = \log \frac{1}{10S} \][/tex]

That’s it! We have derived the equation.

The correct equation that represents the magnitude of an earthquake that is 10 times more intense than a standard earthquake is:

[tex]\[ M = \log \frac{1}{10S} \][/tex]

This matches one of the given options. Therefore, the correct answer is:

[tex]\[ \boxed{M = \log \frac{1}{10 S}} \][/tex]
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