Suppose that a public address system emits sound uniformly in all directions and that there are no reflections. The intensity at a location 20 m away from the sound source is 3.0 × 10-4 W/m2. What is the intensity at a spot that is 87 m away?
To find the intensity at a spot that is 87 m away from the sound source, we can use the inverse square law, which states that the intensity of sound decreases as the square of the distance from the source increases.
The inverse square law equation for sound intensity is:
I₁/I₂ = (r₂/r₁)²
Where:
I₁ is the initial intensity at a distance r₁
I₂ is the unknown intensity at a distance r₂
Let's solve for I₂ using the given information:
I₁ = 3.0 × 10⁻⁴ W/m²
r₁ = 20 m
r₂ = 87 m
Substituting the values into the equation, we have:
I₁/I₂ = (r₂/r₁)²
Using cross multiplication, we can solve for I₂:
I₂ = I₁ * (r₁/r₂)²
Calculating the intensity at 87 m away:
I₂ = 3.0 × 10⁻⁴ W/m² * (20 m/87 m)²
I₂ = 3.0 × 10⁻⁴ W/m² * (0.2299)²
I₂ ≈ 1.058 × 10⁻⁵ W/m²
Therefore, the intensity at a spot 87 m away from the sound source is approximately 1.058 × 10⁻⁵ W/m².