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The loudness, [tex]L[/tex], measured in decibels ([tex]Db[/tex]), of a sound intensity, [tex]I[/tex], measured in watts per square meter, is defined as

[tex]\[ L = 10 \log \frac{I}{I_0} \][/tex]

where [tex]I_0 = 10^{-12}[/tex] is the least intense sound a human ear can hear.

What is the approximate loudness of a rock concert with a sound intensity of [tex]10^{-1}[/tex]?

A. 2 [tex]Db[/tex]
B. 22 [tex]Db[/tex]
C. 60 [tex]Db[/tex]
D. 110 [tex]Db[/tex]


Sagot :

To calculate the loudness, [tex]\(L\)[/tex], of a sound intensity of [tex]\(l = 10^{-1}\)[/tex] watts per square meter using the formula [tex]\(L = 10 \log \frac{l}{I_0}\)[/tex], where [tex]\(I_0 = 10^{-12}\)[/tex] W/m[tex]\(^2\)[/tex] is the reference intensity level (the least intense sound a human ear can hear), we follow these steps:

1. Identify the given values:
- The sound intensity, [tex]\( l = 10^{-1} \)[/tex] W/m[tex]\(^2\)[/tex]
- The reference intensity level, [tex]\( I_0 = 10^{-12} \)[/tex] W/m[tex]\(^2\)[/tex]

2. Substitute the values into the formula:

[tex]\[ L = 10 \log \left( \frac{10^{-1}}{10^{-12}} \right) \][/tex]

3. Simplify the fraction inside the logarithm:

[tex]\[ \frac{10^{-1}}{10^{-12}} = 10^{-1 - (-12)} = 10^{-1 + 12} = 10^{11} \][/tex]

4. Take the logarithm base 10 of [tex]\(10^{11}\)[/tex]:

[tex]\[ \log (10^{11}) = 11 \][/tex]

5. Multiply by 10 to convert to decibels:

[tex]\[ L = 10 \times 11 = 110 \text{ dB} \][/tex]

Therefore, the approximate loudness of a rock concert with a sound intensity of [tex]\(10^{-1}\)[/tex] W/m[tex]\(^2\)[/tex] is [tex]\(110 \text{ dB}\)[/tex].

The correct answer is:
[tex]\[ 110 \text{ dB} \][/tex]