To find the decibel (dB) value of the noise on the construction site, we will use the formula:
[tex]\[ dB = 10 \log I \][/tex]
where [tex]\( I \)[/tex] is the intensity ratio of the noise to the human hearing threshold. In this case, the noise is 100,000 times greater than the human hearing threshold. Thus, [tex]\( I = 100,000 \)[/tex].
Let's go through the steps to calculate the decibel value:
1. Identify the intensity ratio [tex]\( I \)[/tex]:
Given, [tex]\( I = 100,000 \)[/tex].
2. Apply the decibel formula:
Substitute [tex]\( I \)[/tex] into the formula [tex]\( dB = 10 \log I \)[/tex]:
[tex]\[ dB = 10 \log 100,000 \][/tex]
3. Calculate the logarithm:
To find the logarithm of 100,000, recognize that 100,000 can be written as [tex]\( 10^5 \)[/tex]:
[tex]\[ \log 100,000 = \log (10^5) \][/tex]
Using the property of logarithms [tex]\(\log (10^b) = b \log 10\)[/tex], and knowing that [tex]\(\log 10\)[/tex] is 1:
[tex]\[ \log (10^5) = 5 \log 10 = 5 \][/tex]
Thus,
[tex]\[ \log 100,000 = 5 \][/tex]
4. Substitute the logarithm value back into the decibel formula:
[tex]\[ dB = 10 \times 5 = 50 \][/tex]
5. Round the result, if needed:
Since the result is already a whole number, there is no need to round further.
Therefore, the decibel value of the noise is [tex]\( 50 \)[/tex].
