The intensity \(I\) of a sound wave is related to its amplitude \(A\) by the equation:
\[ I \propto A^2 \]
This means that if the intensity of the wave increases by a factor, the amplitude must change by the square root of that factor.
Given that the intensity increases from \(I_0\) to \(16I_0\), we can set up the following relationship:
\[ \frac{I_f}{I_0} = 16 \Rightarrow \frac{A_f^2}{A_o^2} = 16 \Rightarrow \frac{A_f}{A_o} = 4 \]
Thus, the amplitude must increase by a factor of 4. If the original amplitude is \(A_o = x_o\), then the required amplitude is:
\[ A_f = 4x_o \]
Next, we also need to consider the distance from the source. The intensity of a sound wave decreases with distance from the source according to the inverse square law:
\[ I \propto \frac{1}{r^2} \]
Since the intensity is to be \(16I_0\), we can express the new distance \(r_f\) in terms of the original distance \(r_o = 4.0\) m. The intensity change can be summarized as follows:
\[ \frac{I_f}{I_o} = \frac{16I_0}{I_0} = 16 \Rightarrow \frac{r_o^2}{r_f^2} = 16 \Rightarrow \frac{4.0^2}{r_f^2} = 16 \Rightarrow r_f^2 = \frac{16}{16} = 1 \Rightarrow r_f = 1.0 \text{ m} \]
Now, substituting this distance with the factors calculated for amplitude:
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First Option: Increase the amplitude to \(8x_o\) and stand 2 meters from the source.
- Amplitude: \(8x_o\) (Not correct; should be \(4x_o\))
- Distance: 2 m (Not correct)
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Second Option: Increase the amplitude to \(4x_o\) and stand 10 meters from the source.
- Amplitude: \(4x_o\) (Correct)
- Distance: 10 m (Not correct; intensity will decrease)
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Third Option: Increase the amplitude to \(2x_o\) and stand 2 meters from the source.
- Amplitude: \(2x_o\) (Not correct)
- Distance: 2 m (Not correct)
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Fourth Option: Increase the amplitude to \(4x_o\) and stand 8 meters from the source.
- Amplitude: \(4x_o\) (Correct)
- Distance: 8 m (Correct, the intensity will not reach \(16I_0\) but is still close, but not \(4.0\) m hence not applicable)
Conclusion: Based on all calculations, none of the options provided will fit exactly based on theoretical values thus the second option is the best fit regarding the amplitude requirement:
Increase the amplitude to \(4x_o\) and stand 10 meters from the source.
1 answer