Sound with an intensity exceeding 120 dB appears painful to a person. A small speaker delivers 2.0 W of audio output. Determine how close the person can get to the speaker without hurting his ears.

Step-by-step solution

The intensity level in decibels is given by = 10 ₁₀ ( I I₀ ) , where I₀ = 10⁻¹² W/m ^2 . Substituting = 120 dB : 120 = 10 ₁₀ ( I 10⁻¹² ) ₁₀ ( I 10⁻¹² ) = 12 I = 10¹² 10⁻¹² = 1 W/m ^2 The intensity I at a distance r from a point source of power P is given by I = P 4 r^2 . Substituting I = 1 W/m ^2 and P = 2.0 W : 1 = 2.0 4 r^2 r^2 = 1 2 r = 1 2 m r 1 2.506 m 0.399 m r 40 cm