Certainly! Let's solve this problem step by step:
1. Understanding the Problem:
- Initial distance from the engine: [tex]\(8\)[/tex] meters.
- Initial sound intensity at that distance: [tex]\(20 \frac{w}{m^2}\)[/tex].
- New distance from the engine: [tex]\(4\)[/tex] meters.
- We need to find the new sound intensity at the new distance.
2. Concept to be Applied:
- The problem requires the use of the inverse square law for sound intensity. This law states that the intensity of sound is inversely proportional to the square of the distance from the source. Mathematically, this can be written as:
[tex]\[
I \propto \frac{1}{d^2}
\][/tex]
- If we have an initial intensity [tex]\(I_1\)[/tex] at a distance [tex]\(d_1\)[/tex], and we move to a new distance [tex]\(d_2\)[/tex], the new intensity [tex]\(I_2\)[/tex] can be found using the formula:
[tex]\[
I_2 = I_1 \left( \frac{d_1}{d_2} \right)^2
\][/tex]
3. Given Data:
- [tex]\(d_1 = 8\)[/tex] meters.
- [tex]\(I_1 = 20 \frac{w}{m^2}\)[/tex].
- [tex]\(d_2 = 4\)[/tex] meters.
4. Applying the Formula:
- Substitute the given data into the inverse square law formula:
[tex]\[
I_2 = 20 \frac{w}{m^2} \left( \frac{8 \text{ meters}}{4 \text{ meters}} \right)^2
\][/tex]
- Simplify the fraction inside the brackets:
[tex]\[
\left( \frac{8}{4} \right)^2 = 2^2 = 4
\][/tex]
- Multiply the initial intensity by the result:
[tex]\[
I_2 = 20 \frac{w}{m^2} \times 4 = 80 \frac{w}{m^2}
\][/tex]
5. Conclusion:
- The new sound intensity at a distance of 4 meters from the engine is [tex]\(80 \frac{w}{m^2}\)[/tex].
Thus, the measured sound intensity at the new distance is [tex]\(80 \frac{w}{m^2}\)[/tex].
