Question

The intensity level of a "Super-Silent" power lawn mower at a distance of 1.0 m is 100 dB. You wake up one morning to find that four of your neighbors are all mowing their lawns using identical "Super-Silent" mowers. When they are each 20 m from your open bedroom window, what is the intensity level of the sound in your bedroom? You can neglect any absorption, reflection, or interference of the sound. The lowest detectable intensity is 1.0 × 10-12 W/m2.

Answer 1

The intensity level of the sound in the bedroom when 4 neighbors are mowing their lawns at a distance of 20 m can be calculated using the inverse square law and the formula for intensity level conversion. The initial intensity level of the lawn mower at 1.0 m can be converted to watts per meter squared using the intensity level formula. Then, the new intensity level can be calculated by applying the inverse square law to account for the increased distance.

Step-by-step explanation:

The intensity level of a sound wave can be calculated using the formula:

IL = 10 log(I/I₀)

Where IL is the intensity level in decibels (dB), I is the intensity of the sound wave in watts per meter squared, and I₀ is the reference intensity of 10⁻¹² W/m². Using this formula, we can convert the intensity level of the 'Super-Silent' lawn mower at 1.0 m distance to watts per meter squared:

IL₁ = 100 dB

I₁ = I₀ * 10^(IL₁/10) = (10⁻¹²)(10^(100/10))

Now, to find the intensity level of the sound in the bedroom when the 4 neighbors are each 20 m away from the open window, we can use the inverse square law:

IL₂ = IL₁ + 20 log(d/d₀)

Where d is the new distance (20 m), and d₀ is the initial distance (1.0 m). Using this formula, we can calculate the new intensity level:

IL₂ = 100 + 20 log(20/1.0)