Sound with intensity larger than 120 dB appears painful to a person. A small speaker delivers 2.0 W of audio output. How close can the person get to the speaker without hurting his ears ?

Explanation

To solve the problem step-by-step, we will follow the outlined approach in the video transcript.

### Step 1: Understand the given information
We are given:
- Sound intensity level (β) = 120 dB
- Power of the speaker (P) = 2.0 W
- Reference intensity (I₀) = 10⁻¹² W/m²

### Step 2: Use the formula for sound intensity level
The formula for sound intensity level in decibels is:
\[ \beta = 10 \log_{10} \left(\frac{I}{I_0}\right) \]
where \(I\) is the intensity of the sound.

### Step 3: Rearrange the formula to find intensity (I)
We can rearrange the formula to solve for \(I\):
\[ \beta = 10 \log_{10} \left(\frac{I}{10^{-12}}\right) \]
Dividing both sides by 10 gives:
\[ \frac{\beta}{10} = \log_{10} \left(\frac{I}{10^{-12}}\right) \]
Taking the antilogarithm:
\[ 10^{\frac{\beta}{10}} = \frac{I}{10^{-12}} \]
Thus,
\[ I = 10^{\frac{\beta}{10}} \times 10^{-12} \]

### Step 4: Substitute the values to find I
Substituting \(\beta = 120\) dB:
\[ I = 10^{\frac{120}{10}} \times 10^{-12} = 10^{12} \times 10^{-12} = 1 \text{ W/m}^2 \]

### Step 5: Relate intensity to power and area
The intensity \(I\) is also related to power and area by:
\[ I = \frac{P}{A} \]
where \(A\) is the area of a sphere given by \(A = 4\pi R^2\). Thus,
\[ I = \frac{P}{4\pi R^2} \]

### Step 6: Substitute I and P into the equation
We can substitute \(I = 1\) W/m² and \(P = 2\) W into the equation:
\[ 1 = \frac{2}{4\pi R^2} \]

### Step 7: Solve for R
Rearranging gives:
\[ 4\pi R^2 = 2 \]
\[ R^2 = \frac{2}{4\pi} = \frac{1}{2\pi} \]
Taking the square root:
\[ R = \sqrt{\frac{1}{2\pi}} = \frac{1}{\sqrt{2\pi}} \]

### Step 8: Calculate the numerical value of R
Calculating \(R\):
\[ R \approx \sqrt{0.159} \approx 0.399 \text{ m} \]

### Step 9: Convert to centimeters
Converting meters to centimeters:
\[ R \approx 0.399 \text{ m} \approx 39.9 \text{ cm} \approx 40 \text{ cm} \]

### Final Answer
The person can get as close as **40 centimeters** from the speaker without hurting his ears.

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