An observer is 8 m from a sound source. To what distance should she move so that the sound intensity is halved.

Question

BlackTom AI · Accepted Answer

Question restatement: An observer is 8 m from a sound source. To what distance should she move so that the sound intensity is halved? Answer options: a) 14.1 m, b) 15 m, c) 12 m, d) 11.3 m.

Step-by-step analysis:
- The intensity I of sound from a point source decreases with the square of the distance r from the source: I ∝ 1/r^2. When the intensity is halved, I2 = I1/2.
- If I ∝ 1/r^2, then I1/I2 = (r2^2)/(r1^2). Setting I2 = I1/2 gives (r2^2)/(r1^2) = 2, so r2 = r1 * sqrt(2).
- With the observer initially at r1 = 8 m, the final distance from the source after halving the intensity should be r2 = 8 * sqrt(2) ≈ 8 * 1.414 ≈ 11.313 m.
- Now evaluate each option:
  - Option a: 14.1 m. This would yield a final distance greater than 11.3 m, and since 14.1 m is not the calculated r2, this option would not produce exactly halved intensity according to the inverse-square law. It overestimates the needed distance.
  - Option b: 15 m. Similar to 14.1 m, it implies a larger distance than required, giving a smaller intensity than half, so this is not correct.
  - Option c: 12 m. This is close to 11.3 m but still does not match the precise r2 = 11.313 m. If we treat 12 m as the final distance, the intensity would be I = I1/(12^2) relative to I1/(8^2), which is 64/144 ≈ 0.444, not 0.5, so this option is not exact.
  - Option d: 11.3 m. This matches the calculated r2 = 8 * sqrt(2) ≈ 11.313 m very closely, aligning with the halving condition I ∝ 1/r^2.

- Important nuance: The distance the observer must move from the original position is r2 − r1 ≈ 11.313 − 8 ≈ 3.313 m. However, none of the options present this 3.3 m value; the options instead express the final distance from the source, among which 11.3 m is the correct final distance to achieve half the original intensity.

Summary: Considering the inverse-square relationship, the distance from the source should be about 11.3 m to reduce the intensity by half. Among the options, 11.3 m (option d) is the one that corresponds to the required final distance; the other options do not yield exactly half the original intensity. The exact movement distance from the initial position would be approximately 3.3 m, which is not listed among the choices.

An observer is 8 m from a sound source. To what distance should she move so that the sound intensity is halved.

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