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A middle-aged man typically has poorer hearing than a middle-aged woman. In one case a woman can just begin to hear a musical tone, while a man can just begin to hear the tone only when its intensity level in increased by 6.0 dB relative to that for the woman. What is the ratio of the sound intensity just detected by the man to that just detected by the woman?

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Answer:

4

Explanation:

[tex]I_0[/tex] = Threshold intensity for woman

I = Intensity of sound for man

[tex]\beta[/tex] = Intensity level of sound = 6 dB

Intensity level of sound is given by

[tex]\beta=10log\dfrac{I}{I_0}\\\Rightarrow 6=10log\dfrac{I}{I_0}\\\Rightarrow \dfrac{6}{10}=log\dfrac{I}{I_0}\\\Rightarrow 10^{\dfrac{6}{10}}=\dfrac{I}{I_0}\\\Rightarrow \dfrac{I}{I_0}=10^{\dfrac{6}{10}}\\\Rightarrow \dfrac{I}{I_0}=10^{0.6}\\\Rightarrow \dfrac{I}{I_0}=3.98\approx 4[/tex]

The ratio is 4

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