Final answer:
To calculate the power output of a speaker with a sound intensity level of 112 dB at 20 m, the formula “IL(dB) = 10 × log(I / I0)” is used. After finding the intensity at 20 m, you apply the inverse square law (P = I × 4πr²) to calculate power output. For the distance where the intensity is 0.1 W/m², the distance r is calculated from the output power.
Step-by-step explanation:
Calculating Speaker's Power Output Based on Sound Intensity Level
To find the speaker's power output when the sound intensity level is 112 dB at a distance of 20 m, we use the formula for sound intensity level (IL) in decibels (dB):
IL(dB) = 10 × log(I / I0)
Where I is the intensity in watts per meter squared (W/m²) and I0 is the reference intensity, which is typically 1 × 10^-12 W/m² for air.
First, we convert the intensity level from dB to intensity in W/m²:
I = I0 × 10^(IL/10)
After calculating the intensity at 20 m, we use the fact that intensity decreases with the square of the distance from the source (Inverse Square Law) to find the total power output (P) of the speaker, as it would be the same at any distance from the speaker:
P = I × 4πr²
Here, r is the distance from the speaker (20 m in this case), and the factor 4πr² represents the surface area of a sphere with radius r.
Following the same principle, we can find the distance at which the intensity becomes 0.1 W/m² by solving for r using the inverse of the sound intensity formula:
r = √(P / (I × 4π))