Most people interpret a `9.0 dB` increase in sound intensity level as a doubling in loudness. By what factor must the sound intensity be increase to double the loudness?

Options

A. `1 m//s`
B. `2 m//s`
C. `3 m//s`
D. `4 m//s`

Explanation

To determine the factor by which sound intensity must be increased to double the perceived loudness, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship Between Loudness and Intensity:**
Loudness (L) is measured in decibels (dB), and an increase of 9.0 dB is generally perceived as a doubling of loudness. We can express this relationship using the formula for sound intensity level in decibels:
\[ L = 10 \log_{10} \left(\frac{I}{I_0}\right) \]
where \(I\) is the intensity of the sound and \(I_0\) is the reference intensity.

2. **Setting Up the Equations:**
Let \(I_1\) be the initial intensity and \(I_2\) be the intensity after the increase. The initial loudness can be expressed as:
\[ L_1 = 10 \log_{10} \left(\frac{I_1}{I_0}\right) \]
After doubling the loudness, we have:
\[ L_2 = 10 \log_{10} \left(\frac{I_2}{I_0}\right) \]
Given that \(L_2 = L_1 + 9.0\) dB, we can express this as:
\[ L_2 = L_1 + 9.0 \]

3. **Subtracting the Two Equations:**
We can subtract the first equation from the second:
\[ L_2 - L_1 = 10 \log_{10} \left(\frac{I_2}{I_0}\right) - 10 \log_{10} \left(\frac{I_1}{I_0}\right) \]
This simplifies to:
\[ 9.0 = 10 \left( \log_{10} \left(\frac{I_2}{I_0}\right) - \log_{10} \left(\frac{I_1}{I_0}\right) \right) \]
Using the property of logarithms, we can combine the logs:
\[ 9.0 = 10 \log_{10} \left(\frac{I_2}{I_1}\right) \]

4. **Solving for the Intensity Ratio:**
Dividing both sides by 10 gives:
\[ 0.9 = \log_{10} \left(\frac{I_2}{I_1}\right) \]
Converting from logarithmic to exponential form:
\[ \frac{I_2}{I_1} = 10^{0.9} \]

5. **Calculating the Factor:**
Therefore, the factor by which the intensity must increase to double the loudness is:
\[ I_2 = 10^{0.9} I_1 \]
This means the intensity must increase by a factor of \(10^{0.9}\).

### Final Answer:
The sound intensity must increase by a factor of \(10^{0.9}\) to double the loudness.

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