Question 1192174
**Understanding Sound Intensity and Decibels**

Sound intensity is measured on a logarithmic scale using decibels (dB). This scale is based on the equation:

β = 10 log (I / I₀)

where:

* β is the sound intensity level in decibels (dB)
* I is the sound intensity in watts per square meter (W/m²)
* I₀ is the reference intensity, usually the threshold of hearing (1.0 x 10⁻¹² W/m²)

**Calculations**

**a) Threshold of Hearing**

The threshold of hearing is our reference point (I₀ = 1.0 x 10⁻¹² W/m²). Plugging this into the equation:

β = 10 log (1.0 x 10⁻¹² / 1.0 x 10⁻¹²) = 10 log (1) = 0 dB

**The intensity level of the threshold of hearing is 0 dB.**

**b) Whisper**

A whisper has an intensity of 1.0 x 10⁻¹⁰ W/m².  Calculating the decibel level:

β = 10 log (1.0 x 10⁻¹⁰ / 1.0 x 10⁻¹²) = 10 log (100) = 20 dB

**The intensity level of a whisper is 20 dB.**

**c) Siren**

A siren at 30 meters has an intensity of 1.0 x 10⁻² W/m².  Calculating the decibel level:

β = 10 log (1.0 x 10⁻² / 1.0 x 10⁻¹²) = 10 log (10¹⁰) = 100 dB

**The intensity level of the siren is 100 dB.**

**Comparison**

To find how much louder the siren is compared to the whisper, we subtract their decibel levels:

Difference = 100 dB - 20 dB = 80 dB

**The siren sounds 80 dB louder than the whisper.**