In a demonstration of the perception of loudness, a teacher sets up a loudspeaker (P) on a stand at the centre of the school oval.The loudspeaker emits sound equally in all directions.At the point X, 10 m from the loudspeaker, a student, measures the sound intensity level to be [math: Lo dB] L_o \ dB The student now moves to point Y, 30 m away from the loudspeaker.Which of the sound intensity levels listed below (A - D) is the sound intensity level at point Y?

Question

BlackTom AI · Accepted Answer

First, let's restate the scenario and what is being asked: a point source loudspeaker radiates sound uniformly in all directions. At distance 10 m (point X) the sound intensity level is Lo dB. The listener moves to 30 m (point Y). We are to determine the sound intensity level at Y.

Option a: Lo − 4.8 dB. This suggests a decrease of 4.8 dB when moving from 10 m to 30 m. To verify, recall that for a point source, intensity I falls off as 1/r^2, so the level decreases by 10 log10(r2^2/r1^2) = 20 log10(r2/r1). With r2/r1 = 30/10 = 3, the drop is 20 log10(3) ≈ 20 × 0.477 = 9.54 dB. So the correct drop is about 9.5 dB, not 4.8 dB. Therefore option a is incorrect.

Option b: Lo/9 dB. This format implies a unit of decibels with a fractional Lo divided by 9, which is not how decibel levels behave. Decibel levels change additively by the logarithm of intensity ratios, not by dividing Lo itself. So this cannot be correct on physical grounds. Hence option b is incorrect.

Option c: Lo/3 dB. Similar to option b, this incorrectly treats the level as a simple division of Lo by 3 in dB units. The correct approach uses a logarithmic difference, not direct division of the reference level. Thus option c is incorrect.

Option d: Lo − 9.5 dB. This matches the standard 1/r^2 falloff for a point source: at three times farther distance (30 m vs 10 m), the intensity drops by a factor of 9, corresponding to a decrease of 10 log10(9) ≈ 9.54 dB, often approximated as 9.5 dB. Therefore the sound intensity level at Y is Lo − 9.5 dB, making this the correct choice.

In summary, the only option consistent with the inverse-square law for a point source and the resulting 9.5 dB drop in level is option d. The other options fail because they misapply the logarithmic relation or treat decibel levels as linear quantities rather than logarithmic ratios.

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