To determine the approximate intensity, [tex]\( I \)[/tex], of an earthquake with a Richter scale magnitude of 4.8, given that the reference intensity, [tex]\( I_0 \)[/tex], is 1, we can use the Richter scale formula.
The formula for the Richter scale magnitude [tex]\( M \)[/tex] is:
[tex]\[
M = \log_{10} \left( \frac{I}{I_0} \right)
\][/tex]
We are given:
- [tex]\( M = 4.8 \)[/tex]
- [tex]\( I_0 = 1 \)[/tex]
First, we need to isolate [tex]\( I \)[/tex] in this equation. Starting from the given formula:
[tex]\[
4.8 = \log_{10} \left( \frac{I}{I_0} \right)
\][/tex]
Since [tex]\( I_0 = 1 \)[/tex], this simplifies to:
[tex]\[
4.8 = \log_{10} (I)
\][/tex]
To solve for [tex]\( I \)[/tex], we need to rewrite the logarithmic equation in its exponential form. The equation [tex]\( \log_{10} (I) = 4.8 \)[/tex] can be rewritten as:
[tex]\[
I = 10^{4.8}
\][/tex]
Now we calculate [tex]\( 10^{4.8} \)[/tex]:
[tex]\[
I \approx 10^{4.8} \approx 63095.7344480193
\][/tex]
Thus, the intensity [tex]\( I \)[/tex] of the earthquake is approximately 63,095. Therefore, the correct answer is:
C. 63,096
