The problem states that the loudness $L$ of a sound in decibels (dB) can be calculated using the formula $L = 10 \log(\frac{I}{I_0})$, where $I$ is the intensity of the sound in watts per square meter ($\frac{W}{m^2}$) and $I_0 = 10^{-12} \frac{W}{m^2}$. We need to determine the intensity of the sound for an audience applauding if the sound level is 100 dB.

1. Problem Description

The problem states that the loudness $L$ of a sound in decibels (dB) can be calculated using the formula $L = 10 \log(\frac{I}{I_0})$, where $I$ is the intensity of the sound in watts per square meter ($\frac{W}{m^2}$) and $I_0 = 10^{-12} \frac{W}{m^2}$. We need to determine the intensity of the sound for an audience applauding if the sound level is 100 dB.

2. Solution Steps

We are given the formula:

$L = 10 \log(\frac{I}{I_0})$

We are given $L = 100$ dB and $I_0 = 10^{-12} \frac{W}{m^2}$. We want to find $I$.

Substituting the given values into the formula, we get:

$100 = 10 \log(\frac{I}{10^{-12}})$

Divide both sides by 10:

$10 = \log(\frac{I}{10^{-12}})$

Since the logarithm is base 10, we can rewrite the equation as:

$10^{10} = \frac{I}{10^{-12}}$

Multiply both sides by $10^{-12}$:

$I = 10^{10} \times 10^{-12}$

$I = 10^{10-12}$

$I = 10^{-2}$

$I = \frac{1}{10^2}$

$I = \frac{1}{100}$

Therefore, $I = \frac{1}{100} \frac{W}{m^2}$.

We can also express this as $I=0.01 \frac{W}{m^2}$.

The question gave us $\frac{1}{1000}$ as an option, but our answer is $\frac{1}{100}$.

The given options seem to be incorrect as the value closest to our answer is 100, but this is incorrect and refers to the Decibel level.

3. Final Answer

$\frac{1}{100}$

However, since $\frac{1}{100}$ is not an option, and since the question states the sound level is 100 dB, we solve the problem assuming the question means to state "if the log of the ratio of the intensity to the reference intensity is 100".

$100 = \log_{10}{\frac{I}{I_0}}$

$10^{100} = \frac{I}{I_0}$

$I = 10^{100} \times I_0$

$I = 10^{100} \times 10^{-12}$

$I = 10^{88}$

However, this seems physically improbable.

Reviewing the other possible options, it seems most likely that there is an error, and the correct answer, derived above, should be 1/100, which is $0.01$.

With the available options being:

1000, 20, 1/1000, 100

None are correct. The closest is likely 100, however the calculation showed that 100 is the Decibel level and not the intensity of the sound.

Final Answer: 1/100