Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m^2 at a distance of 4.3 m from the source. (a) What is the intensity at a distance of 3.1 m from the source?

Verified step by step guidance

1. Identify that the problem involves the inverse square law, which states that the intensity of a wave is inversely proportional to the square of the distance from the source. The formula for this relationship is $I_1 = I_2 \left(\frac{d_2}{d_1}\right)^2$, where $I_1$ and $I_2$ are the intensities at distances $d_1$ and $d_2$ respectively.
2. Substitute the given values into the formula. Here, $I_2 = 0.026 \, \text{W/m}^2$ (intensity at 4.3 m), $d_2 = 4.3 \, \text{m}$ (distance at which intensity is given), and $d_1 = 3.1 \, \text{m}$ (distance at which intensity needs to be found).
3. Rearrange the formula to solve for $I_1$: $I_1 = 0.026 \, \text{W/m}^2 \left(\frac{4.3 \, \text{m}}{3.1 \, \text{m}}\right)^2$.
4. Calculate the value inside the parentheses first, which is the ratio of the distances squared.
5. Finally, multiply the initial intensity $I_2$ by the square of the ratio of distances to find the intensity $I_1$ at 3.1 m.