Final answer:
The maximum distance at which a tornado siren can be heard is calculated using the inverse square law, solving for the distance r where the intensity drops to the threshold of hearing. This requires finding the power of the siren at a given distance and then determining the new distance at which the intensity meets the weakest intensity that can be heard over background noise.
Step-by-step explanation:
The student is asking how to estimate the maximum distance at which a tornado siren can be heard, given its intensity at a certain distance and the weakest intensity likely to be heard over background noise. Sound intensity decreases with the square of the distance from the source based on the inverse square law. Starting from an intensity of 0.120 W/m² at a distance of 57.0 m, we can use the formula for intensity (I = P/4πr², where P is the power of the source and r the distance) to determine the distance at which the intensity would drop to the threshold intensity of 1μW/m² (1 x 10^-6 W/m²).
To find the maximum distance:
- Assume that the power P of the siren is constant.
- Use the given intensity at 57.0 m to calculate P.
- Use the calculated power to find the new radius r when intensity I equals 1μW/m².
Let's calculate the power P based on the given intensity at 57.0 m: P = I * 4πr² = 0.120 W/m² * 4π * (57.0 m)²
Next, use this power to find the new distance r where I = 1μW/m²: 1 x 10^-6 W/m² = P / (4πr²)
Solve for r to find the maximum distance: r = √(P / (4π * 1 x 10^-6 W/m²))
By substituting the value of P found earlier, the student will get the maximum distance at which the siren can be heard over the background noise.