Question

For a sound with intensity I (in watts per square meter), the loudness L(1) of the

sound (in decibels) is given by the function L(J) = 10 log I/I0
To Where I0 is the
intensity of a barely audible sound (about 10-12 watts per square meter). The sound of a coach's whistle is five times greater than the intensity of the referee's whistle. Find the difference in the decibel levels of the sounds made by the coach and the referee.

Answer 1

Let I_referee be the intensity of the referee's whistle and I_coach be the intensity of the coach's whistle. We know that I_coach = 5 * I_referee.

The loudness of the referee's whistle L_referee is given by:

L_referee = 10 * log(I_referee / I0)

The loudness of the coach's whistle L_coach is given by:

L_coach = 10 * log(I_coach / I0) = 10 * log(5 * I_referee / I0)

Using the logarithmic identity log(a*b) = log(a) + log(b), we can simplify this expression:

L_coach = 10 * (log(5) + log(I_referee / I0))

L_coach = 10 * log(5) + 10 * log(I_referee / I0)

L_coach = 10 * log(5) + L_referee

Therefore, the difference in loudness between the coach and referee's whistle is:

L_coach - L_referee = 10 * log(5)

Using a calculator, we can evaluate this to be approximately 7.96 decibels (dB).

Therefore, the difference in decibel levels of the sounds made by the coach and the referee is approximately 7.96 dB.