To find the sound intensity experienced by the other student, we can use the inverse square law of sound propagation. According to this law, the intensity of sound decreases proportionally to the square of the distance from the source. Here’s a detailed step-by-step solution to the problem:
1. Given Data:
- Initial sound intensity [tex]\( I_1 = 1.1 \times 10^{-7} \, W/m^2 \)[/tex]
- Initial distance from the source [tex]\( r_1 = 3.0 \, m \)[/tex]
- Distance of the other student from the source [tex]\( r_2 = 4.0 \, m \)[/tex]
2. Inverse Square Law Formula:
The inverse square law states that:
[tex]\[
I_2 = I_1 \left(\frac{r_1}{r_2}\right)^2
\][/tex]
Where [tex]\( I_2 \)[/tex] is the intensity at distance [tex]\( r_2 \)[/tex].
3. Substitute the Given Values into the Formula:
[tex]\[
I_2 = 1.1 \times 10^{-7} \left(\frac{3.0}{4.0}\right)^2
\][/tex]
4. Simplify the Fraction:
[tex]\[
\left(\frac{3.0}{4.0}\right)^2 = \left(\frac{3}{4}\right)^2 = 0.75^2 = 0.5625
\][/tex]
5. Calculate the New Intensity:
[tex]\[
I_2 = 1.1 \times 10^{-7} \times 0.5625 = 6.1875 \times 10^{-8} \, W/m^2
\][/tex]
6. Determine the Closest Answer:
The calculated intensity [tex]\( 6.1875 \times 10^{-8} \, W/m^2 \)[/tex] matches the closest to one of the given options:
- [tex]\( 8.3 \times 10^{-8} W/m^2 \)[/tex]
- [tex]\( 2.5 \times 10^{-8} W/m^2 \)[/tex]
- [tex]\( 7.8 \times 10^{-7} W/m^2 \)[/tex]
- [tex]\( 6.2 \times 10^{-8} W/m^2 \)[/tex] (this one)
- [tex]\( 1.5 \times 10^{-7} W/m^2 \)[/tex]
Thus, the reasonable estimate for the sound intensity that the other student measures is [tex]\( 6.2 \times 10^{-8} W / m^2 \)[/tex].
