To understand how loud a sound is compared to the threshold of hearing, we use the concept of decibels (dB), which is a logarithmic unit that measures the intensity of a sound.
The formula to calculate the decibel level of a sound is:
[tex]\[ \text{decibel (dB)} = 10 \times \log_{10} \left( \frac{I}{I_0} \right) \][/tex]
where:
- [tex]\( I \)[/tex] is the intensity of the sound,
- [tex]\( I_0 \)[/tex] is the reference intensity, often the threshold of hearing.
In this case, the sound in question is 1000 times louder than the threshold of hearing. Therefore, the intensity of the sound [tex]\( I \)[/tex] is 1000 times the reference intensity [tex]\( I_0 \)[/tex].
Putting this into the formula, we get:
[tex]\[ \text{decibel (dB)} = 10 \times \log_{10} \left( \frac{1000 \times I_0}{I_0} \right) \][/tex]
Since [tex]\( I_0 \)[/tex] cancels out, the equation simplifies to:
[tex]\[ \text{decibel (dB)} = 10 \times \log_{10} (1000) \][/tex]
We need to find the logarithm of 1000 (base 10):
[tex]\[ \log_{10} (1000) = 3 \][/tex]
So the equation becomes:
[tex]\[ \text{decibel (dB)} = 10 \times 3 \][/tex]
[tex]\[ \text{decibel (dB)} = 30 \][/tex]
Thus, a sound that is 1000 times louder than the threshold of hearing measures 30 decibels.
Therefore, the correct answer is:
D. 30
