Let's break down the problem step by step to understand how to compute the intensity of sound in decibels given the provided values.
The intensity of a sound in decibels (dB) is given by the formula:
[tex]\[ dB = 10 \log_{10} \left(\frac{I}{I_0} \right) \][/tex]
where:
- [tex]\( I \)[/tex] is the intensity of the given sound.
- [tex]\( I_0 \)[/tex] is the reference threshold of hearing intensity.
- [tex]\( \log_{10} \)[/tex] denotes the base-10 logarithm.
In this problem, we are given that [tex]\( I = 10^{22} \times I_0 \)[/tex].
Let's plug this value into the formula to find the intensity in decibels.
[tex]\[ dB = 10 \log_{10} \left(\frac{10^{22} \cdot I_0}{I_0}\right) \][/tex]
Since [tex]\( I_0 \)[/tex] cancels out in the fraction, we have:
[tex]\[ dB = 10 \log_{10} \left(10^{22}\right) \][/tex]
The property of logarithms tells us that [tex]\(\log_{10} (10^x) = x\)[/tex]. Therefore:
[tex]\[ \log_{10} \left(10^{22}\right) = 22 \][/tex]
Now, substituting this back into our formula, we get:
[tex]\[ dB = 10 \times 22 \][/tex]
[tex]\[ dB = 220 \][/tex]
Thus, the intensity in decibels is:
[tex]\[ |I(dB)| = 220 \][/tex]
Therefore, given the provided options, the correct answer is:
[tex]\[ \boxed{320} \][/tex]
