Two loudspeakers are being compared. One is preceived to be 32 times louder than the other. The difference in intensity levels between the two, when measured in decibles, is

Options

A. 60
B. 40
C. 50
D. 30

Answer

To find the difference in intensity levels between two loudspeakers when one is perceived to be 32 times louder than the other, we can use the formula for sound intensity level in decibels (dB):

\[ L = 10 \log_{10} \left( \frac{I_2}{I_1} \right) \]

Where:
- \(L\) is the difference in intensity level in decibels,
- \(I_1\) is the intensity of the quieter loudspeaker,
- \(I_2\) is the intensity of the louder loudspeaker.

Given that one loudspeaker is perceived to be 32 times louder than the other, we can express this relationship in terms of intensity. The perceived loudness is related to the intensity by the following relationship:

\[ \text{Loudness} \propto I^{\frac{1}{n}} \]

Where \(n\) is typically taken as 10 for sound. Therefore, if one loudspeaker is 32 times louder, we can say:

\[ \frac{I_2}{I_1} = 32 \]

Now, substituting this into the formula for the difference in intensity levels:

\[ L = 10 \log_{10} \left( \frac{I_2}{I_1} \right) = 10 \log_{10} (32) \]

Next, we need to calculate \(\log_{10} (32)\). We can express 32 as \(2^5\):

\[ \log_{10} (32) = \log_{10} (2^5) = 5 \log_{10} (2) \]

Using the approximate value \(\log_{10} (2) \approx 0.301\):

\[ \log_{10} (32) \approx 5 \times 0.301 = 1.505 \]

Now substituting back into the equation for \(L\):

\[ L = 10 \times 1.505 = 15.05 \text{ dB} \]

Thus, the difference in intensity levels between the two loudspeakers is approximately:

\[ L \approx 15.05 \text{ dB} \]

However, since we need to find the difference in intensity levels when one is perceived to be 32 times louder, we can also use the fact that every doubling of intensity corresponds to an increase of about 10 dB.

Since \(32 = 2^5\), we can say that:

\[ L = 10 \times 5 = 50 \text{ dB} \]

Thus, the final answer is:

\[ \text{The difference in intensity levels is } 50 \text{ dB.} \]