(a) By what factor must the sound intensity be increased the sound intensity level by `13.0 dB` ? (b) Explain why you do not need to know the original sound intersity ?

Explanation

To solve the problem step by step, we will use the relationship between sound intensity level (in decibels) and sound intensity.

### Step-by-Step Solution:

**(a)** To find the factor by which the sound intensity must be increased to achieve an increase of 13.0 dB, we start with the formula for sound intensity level:

\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]

where:
- \( L \) is the sound intensity level in decibels (dB),
- \( I \) is the sound intensity,
- \( I_0 \) is the reference sound intensity.

Given that we want to increase the sound intensity level by 13 dB, we can express this as:

\[ L_2 - L_1 = 13 \, \text{dB} \]

Using the formula for sound intensity levels, we have:

\[ L_2 = 10 \log_{10} \left( \frac{I_2}{I_0} \right) \]
\[ L_1 = 10 \log_{10} \left( \frac{I_1}{I_0} \right) \]

Substituting these into our equation gives:

\[ 10 \log_{10} \left( \frac{I_2}{I_0} \right) - 10 \log_{10} \left( \frac{I_1}{I_0} \right) = 13 \]

Factoring out the 10:

\[ 10 \left( \log_{10} \left( \frac{I_2}{I_0} \right) - \log_{10} \left( \frac{I_1}{I_0} \right) \right) = 13 \]

Using the properties of logarithms, we can simplify this to:

\[ 10 \log_{10} \left( \frac{I_2}{I_1} \right) = 13 \]

Dividing both sides by 10:

\[ \log_{10} \left( \frac{I_2}{I_1} \right) = 1.3 \]

To eliminate the logarithm, we exponentiate both sides:

\[ \frac{I_2}{I_1} = 10^{1.3} \]

Calculating \( 10^{1.3} \):

\[ 10^{1.3} \approx 19.95 \]

Rounding this, we find:

\[ \frac{I_2}{I_1} \approx 20 \]

Thus, the sound intensity must be increased by a factor of approximately **20**.

**(b)** We do not need to know the original sound intensity \( I_1 \) because the relationship we derived is independent of its specific value. The equation \( \frac{I_2}{I_1} = 10^{1.3} \) shows that the factor of increase (20) is a ratio. This means that regardless of what \( I_1 \) is, \( I_2 \) will always be 20 times \( I_1 \) to achieve the desired increase in sound intensity level by 13 dB.