Answer:
Phil should stand at a distance of 0.071 times as far as mine to experience the same loudness as me.
Step-by-step 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.