To determine the intensity of the earthquake given its magnitude on the Richter scale, we can use the Richter scale formula for an earthquake's intensity.
The Richter scale formula is:
[tex]\[ M = \log_{10}\left(\frac{I}{I_0}\right) \][/tex]
Where:
- [tex]\( M \)[/tex] is the magnitude on the Richter scale (in this case, 5.6).
- [tex]\( I \)[/tex] is the intensity of the earthquake that we want to find.
- [tex]\( I_0 \)[/tex] is the reference intensity.
We need to isolate [tex]\( I \)[/tex] (the intensity) from this equation:
1. Start by expressing the formula in exponential form:
[tex]\[ 10^M = \frac{I}{I_0} \][/tex]
2. Substitute the given values for [tex]\( M \)[/tex] (5.6) and [tex]\( I_0 \)[/tex] ([tex]\( 7.3 \times 10^9 \)[/tex]):
[tex]\[ 10^{5.6} = \frac{I}{7.3 \times 10^9} \][/tex]
3. To solve for [tex]\( I \)[/tex], multiply both sides of the equation by [tex]\( 7.3 \times 10^9 \)[/tex]:
[tex]\[ I = 7.3 \times 10^9 \times 10^{5.6} \][/tex]
4. Calculate the value of [tex]\( 10^{5.6} \)[/tex]. Note this as a key numerical step:
[tex]\[ 10^{5.6} \simeq 398107.17 \][/tex]
5. Plug this value into the equation:
[tex]\[ I = 7.3 \times 10^9 \times 398107.17 \][/tex]
6. Perform the multiplication to find the intensity [tex]\( I \)[/tex]:
[tex]\[ I \approx 2906182345040527.5 \][/tex]
7. Express this number in scientific notation for clarity:
[tex]\[ I \approx 2.91 \times 10^{15} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{2.91 \times 10^{15}} \][/tex]
The correct option among the given choices is:
E. [tex]\( 2.91 \times 10^{15} \)[/tex]
