A sound source emits a sound wave with a power of 370.7 watts. Calculate the sound intensity at a distance of 8 meters from the source.Give your answer in Wm−2 using standard form to 2 decimal places in the following format without units:[math: 1.22×10−4] should be written as [ 1.22e-4 ]

Question

BlackTom AI · Accepted Answer

Sound intensity from a point source is given by I = P / (4πr^2).

1. Relevant knowledge points and concepts
- Intensity (I) is power per unit area radiated uniformly over a spherical surface: area = 4πr^2.
- Use P = 370.7 W and r = 8.0 m.
- Convert final result to standard (scientific) form with two decimal places in the mantissa as requested.

2. Complete reasoning process step by step
I = P / (4πr^2).
Compute r^2 = 8^2 = 64.
Compute 4πr^2 = 4 × π × 64 = 256π.

3. Calculations with intermediate steps
4πr^2 = 256π ≈ 256 × 3.141592653589793 = 804.247719318987.
I = 370.7 / 804.247719318987 ≈ 0.4609285... (W m−2).

To express in standard form with two decimal places in the mantissa: write 0.4609285 as 4.609285 × 10^−1. Rounding the mantissa to two decimal places gives 4.61 × 10^−1, which in the requested text format is 4.61e-1.

4. Why the correct answer is correct
The formula I = P/(4πr^2) correctly describes the power spread over the surface area of a sphere at distance r. Substituting the given power and distance yields the computed numeric value. Converting to scientific notation and rounding the mantissa to two decimal places yields 4.61e-1, consistent with the numerical value ~0.461 W m−2.

5. Why other common errors would be incorrect
- Omitting the 4π (using area = πr^2) or using r instead of r^2 would give values orders of magnitude off.
- Reporting 0.46 without scientific-format mantissa to two decimal places or using incorrect rounding (e.g., 4.60e-1) would not meet the format/precision requirement.

6. Conclusion and confirmation
The correctly computed sound intensity, formatted as requested, is 4.61e-1, which matches the provided answer.

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