Final answer:
The loudness of the sound at a distance of 0.668 meters from the speaker is calculated using the sound intensity level in decibels formula, resulting in approximately 105.5 dB.
Step-by-step explanation:
To calculate the loudness in decibels (dB) of the sound at a distance of 0.668 meters from the speaker, we can use the formula for sound intensity level (IL) in decibels:
IL(dB) = 10 × log10(I / I0)
Where:
- I is the sound intensity in watts per meter squared (W/m²).
- I0 is the reference intensity, which is considered the threshold of hearing and is 10-12 W/m².
Firstly, we need to calculate the intensity I of the sound at the distance from the speaker:
I = P / (4 × π × r²)
Where:
- P is the acoustical power in watts (0.200 W).
- r is the distance from the speaker in meters (0.668 m).
Plugging these values into the formula gives us:
I = 0.200 W / (4 × π × (0.668 m)2) = 0.200 W / (4 × 3.14159 × 0.446224) = 0.200 W / 5.5911 m² = 0.0358 W/m²
Now, we can calculate the loudness at the distance of 0.668 m:
IL(dB) = 10 × log10(0.0358 W/m² / 10-12 W/m²) = 10 × log10(3.58 × 1010)
This results in a loudness of:
IL(dB) = 10 × (10 + log10(3.58)) = 10 × (10 + 0.5539) = 105.539 dB
Therefore, the loudness of the sound at a distance of 0.668 meters from the speaker is approximately 105.5 dB.