Physics Problem: Sound Intensity Level at Different Distances
Question:
You are 100 m away from the loudspeaker hanging from the dome inside a large indoor stadium. If the sound intensity level is 80 dB, what is the sound intensity level for a person at 60 m away?
Detailed Solution:
Step 1: Understand the Inverse Square Law for Sound Intensity
Sound intensity decreases with the square of the distance from the source. Mathematically:
I ∝ 1 / r²
The ratio of intensities at two distances (r₁ and r₂) is:
I₂ / I₁ = (r₁ / r₂)²
Substituting the given distances:
I₂ / I₁ = (100 / 60)² = (5 / 3)² = 25 / 9 ≈ 2.78
Step 2: Apply the Decibel Formula
The sound level difference in decibels is calculated using:
β₂ − β₁ = 10 × log₁₀(I₂ / I₁)
We know:
- β₁ = 80 dB (at 100 m)
- I₂ / I₁ = 2.78
β₂ − 80 = 10 × log₁₀(2.78) ≈ 10 × 0.444 = 4.44
β₂ = 80 + 4.44 = 84.4 dB
✅ Final Answer:
The sound intensity level at 60 meters is approximately 84.4 dB.
The sound intensity level at 60 meters is approximately 84.4 dB.