Earplugs you can buy in the drugstore have a noise reduction rating of 23 decibels. If you are wearing those earplugs but your friend Phil is not, how far from the speaker should Phil stand to experience the same loudness as you

Phil should stand at a distance of 0.071 times as far as mine to experience the same loudness as me.

Explanation:

From DB = 10logI₂/I₁ where I₁ = intensity at Phil's distance r₁ from the source and I₂ = intensity at my distance distance r₁ from the source. DB = decibel = -23 dB since there is a noise reduction at r₁ from the source.

-23 = 10logI₂/I₁

-23/10 = logI₂/I₁

taking antilogarithm of both sides, we have

10⁻²°³ = I₂/I₁

0.005 =  I₂/I₁

I₁ = I₂/0.005

Thus at r₁, I₁ = P₁/4πr₁² and I₂ = P₂/4πr₁² where P₁ = power received at r₁ by Phil and P₂ = power received at r₁ by me at 23 dB reduction

P₁/4πr₁² = P₂/[4πr₁²(0.005)]

P₁ = P₂/0.005

P₂ = 0.005P₁ = 0.005P where P is the power from the source.

At r₂, at 23 dB reduction and intensity I₃ from intensity I₁ at distance r₁.

So, -23 dB = 10logI₃/I₁

-23/10 = logI₃/I₁

taking antilogarithm of both sides, we have

10⁻²°³ = I₃/I₁

0.005 =  I₃/I₁

I₃ = 0.005I₁ since the power form the source is the same,

P/4πr₂² = 0.005P/4πr₁²

r₂² = 0.005r₁²

taking square-root of both sides, we have

r₂ = (√0.005)r₁

r₂ = 0.071r₁

So, Phil should stand at a distance of 0.071 times as far as mine to experience the same loudness as me.

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