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The magnitude of an earthquake, measured on the Richter scale, is [tex]\log_{10} \left( \frac{I}{I_0} \right)[/tex], where [tex]I[/tex] is the amplitude registered on a seismograph 100 km from the epicenter of the earthquake, and [tex]I_0[/tex] is the amplitude of an earthquake of a certain (small) size.

The largest earthquake struck with a magnitude of 6.8 on the Richter scale. Express this reading in terms of [tex]I_0[/tex].

[tex]I = \square I_0[/tex]

(Round to the nearest whole number as needed.)

Sagot :

GinnyAnsweridn
To solve for [tex]\( I \)[/tex] in terms of [tex]\( I_0 \)[/tex], we can follow these steps:

1. Understand the Richter scale formula:
[tex]\[ \text{Magnitude} = \log_{10} \left(\frac{I}{I_0}\right) \][/tex]
Given, the magnitude is 6.8.

2. Write down the given magnitude equation:
[tex]\[ 6.8 = \log_{10} \left(\frac{I}{I_0}\right) \][/tex]

3. Solve for the ratio [tex]\(\frac{I}{I_0}\)[/tex]:
[tex]\[ \frac{I}{I_0} = 10^{6.8} \][/tex]

4. Calculate [tex]\( 10^{6.8} \)[/tex]:
[tex]\[ 10^{6.8} \approx 6309573.44480193 \][/tex]

5. Express [tex]\( I \)[/tex] in terms of [tex]\( I_0 \)[/tex]:
[tex]\[ I = 6309573.44480193 \times I_0 \][/tex]

6. Round the value to the nearest whole number:
[tex]\[ I \approx 6309573 \times I_0 \][/tex]

Therefore, the amplitude [tex]\( I \)[/tex] can be expressed in terms of [tex]\( I_0 \)[/tex] as:

[tex]\[ I \approx 6309573 \, I_0 \][/tex]

So, the final answer is:
[tex]\[ I \approx 6309573 \, I_0 \][/tex]
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