Two sources of sound are located on the x axis, and each emits power uniformly in all directions. There are no reflections. One source is positioned at the origin and the other at x = +141 m. The source at the origin emits four times as much power as the other source. Where on the x axis are the two sounds equal in intensity? Note that there are two answers.

Question

Two sources of sound are located on the x axis, and each emits power uniformly in all directions. There are no reflections. One source is positioned at the origin and the other at x = +141 m. The source at the origin emits four times as much power as the other source. Where on the x axis are the two sounds equal in intensity? Note that there are two answers.

BlackTom AI · Accepted Answer

This problem asks where along the x-axis the sound levels from two isotropic sources are equal in intensity. First, identify the setup: Source A with power P is at x = 0, and Source B with power P/4 is at x = +141 m. Because intensity from a point source falls off as 1/r^2, the intensity at a point x from Source A is proportional to P/x^2, and the intensity from Source B is proportional to (P/4)/(x−141)^2. We want the contributions from the two sources to be equal, meaning P/x^2 = (P/4)/(x−141)^2. After canceling P, the equation becomes 1/x^2 = (1/4)/(x−141)^2.

Now solve the equation. A straightforward way is to take square roots of both sides (being careful with absolute values): |x−141| = |x|/2.

Consider the possibilities:
- Case 1: x ≥ 0 and x − 141 ≥ 0 (i.e., x ≥ 141). Then x − 141 = x/2. Solving gives 2x − 282 = x → x = 282. This satisfies x ≥ 141, so 282 m is a valid solution.
- Case 2: x ≥ 0 and x − 141 ≤ 0 (i.e., 0 ≤ x ≤ 141). Then −(x−141) = x/2, which simplifies to −x + 141 = x/2 → 282 = 3x → x = 94. This lies within 0 ≤ x ≤ 141, so 94 m is also valid.
- Case 3: x < 0 would lead to a pair of values that do not produce additional valid solutions beyond the two already found, and the algebra confirms the same two positions when evaluating absolute value relations.

Thus, there are two positions where the intensities from the two sources are equal: at x = 94 m and at x = 282 m. Now evaluate each answer choice:

- Option "94m, 282m": Matches exactly the two computed positions, so this pair is correct.
- Option "-37m, 81m": These values do not satisfy the equation |x−141| = |x|/2 for any combination of signs that yields equality of the two contributions; they arise from incorrect algebra or sign handling.
- Option "197m, 303m": Substituting these into the equality condition shows they do not satisfy P/x^2 = (P/4)/(x−141)^2, so they are not valid solutions.
- Option "None of these": Since we found valid solutions, this option is incorrect.
- Option "60m, 183m": These do not solve the equality condition either; they stem from a misapplication of the square-root step or an incorrect sign assignment.

In summary, the correct pair is 94 m and 282 m, corresponding to the two locations along the x-axis where the two isotropic sources contribute equal intensity at a given point.

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