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In 2009, an earthquake hit Costa Rica, registering a 6.1 on the Richter scale. What was the intensity of this earthquake, assuming the reference value was 1?

[tex]\[ \left(R = \log \left(\frac{I}{I_0}\right)\right) \][/tex]

A. 5.1

B. [tex]\[4.46 \times 10^2\][/tex]

C. [tex]\[1.26 \times 10^5\][/tex]

D. [tex]\[1.26 \times 10^6\][/tex]



Answer :

GinnyAnswer
To determine the intensity of the earthquake, we start with the given formula:

[tex]\[ R = \log \left(\frac{I}{I_0}\right) \][/tex]

We are given:
- [tex]\( R = 6.1 \)[/tex]
- [tex]\( I_0 = 1 \)[/tex] (reference value)

Our goal is to find the intensity [tex]\( I \)[/tex].

Step 1: Plug the given values into the formula:

[tex]\[ 6.1 = \log \left(\frac{I}{1}\right) \][/tex]

Step 2: Simplify the equation since [tex]\( I_0 = 1 \)[/tex]:

[tex]\[ 6.1 = \log(I) \][/tex]

Step 3: Rewrite the equation in exponential form to solve for [tex]\( I \)[/tex]:

[tex]\[ I = 10^{6.1} \][/tex]

Step 4: Calculate [tex]\( 10^{6.1} \)[/tex]:

[tex]\[ 10^{6.1} \approx 1258925.411794166 \][/tex]

So, the intensity of the earthquake is approximately [tex]\( 1,258,925.41 \)[/tex].

Step 5: Express the intensity in scientific notation:

[tex]\[ 1258925.41 \approx 1.26 \times 10^6 \][/tex]

Thus, the correct answer is:

D. [tex]\( 1.26 \times 10^6 \)[/tex]