Respuesta :
Answer:
Approximately [tex]9.5 \times 10^{4}\; {\rm m}[/tex] ([tex]95\; {\rm km}[/tex].)
Explanation:
Under the assumptions, energy from the aircraft would be evenly spread across an imaginary spherical surface centered at the aircraft. The fixed amount of power would be spread across a surface area proportional to the square of the radius this imaginary sphere. Hence, the energy intensity would be inversely proportional to the square of the distance from the aircraft.
In this question, it is given that energy intensity is [tex]10\; {\rm W \cdot m^{-2}}[/tex] at [tex]30.0\; {\rm m}[/tex] from the aircraft. The goal is to find the distance [tex]d[/tex] that would reduce the energy intensity to [tex]1.0\; {\rm \mu W \cdot m^{-2}} = 1.0 \times 10^{-6}\; {\rm \mu W\cdot m^{-2}}[/tex].
Because energy intensity is inversely proportional to the distance from the center:
[tex]\displaystyle \frac{10\; {\rm W \cdot m^{-2}}}{1.0 \times 10^{-6}\; {\rm W\cdot m^{-2}}} = \frac{1}{\displaystyle \left(\frac{30\; {\rm m}}{d}\right)^{2}}[/tex].
[tex]\displaystyle \frac{10\; {\rm W \cdot m^{-2}}}{1.0 \times 10^{-6}\; {\rm W\cdot m^{-2}}} = \left(\frac{d}{30\; {\rm m}}\right)^{2}[/tex].
[tex]\displaystyle \frac{d}{30\; {\rm m}} = \sqrt{\frac{10\; {\rm W \cdot m^{-2}}}{1.0 \times 10^{-6}\; {\rm W \cdot m^{-2}}}}[/tex].
[tex]\begin{aligned} d &= (30\; {\rm m}) \, \sqrt{\frac{10\; {\rm W \cdot m^{-2}}}{1.0 \times 10^{-6}\; {\rm W \cdot m^{-2}}}} \\ &\approx 9.5 \times 10^{4}\; {\rm m}\end{aligned}[/tex].
In other words, the energy intensity would be reduced to [tex]1.0 \times 10^{-6}\; {\rm W\cdot m^{-2}}[/tex] at approximately [tex]95\; {\rm km}[/tex] from the aircraft.
Final answer:
To maintain a peaceful sound environment equivalent to the intensity of a normal conversation (1.0 μW/m²), one should live approximately 94.9 km away from the airport runway, as calculated using the inverse square law for sound intensity.
Explanation:
The student is asking how far they should live from an airport runway to only experience the sound intensity of normal conversation (1.0 μW/m²), assuming the sound of a jet at takeoff is 10 W/m² at a distance of 30 m. To solve this problem, we can use the inverse square law for sound which states that the intensity of sound is inversely proportional to the square of the distance from the source. Given the intensity of sound decreases with the square of the distance, we can set up a ratio comparing the intensities and distances:
I1/I2 = (D2/D1)²
e 1 is the intensity of the jet plane at 30 m (10 W/m²), I2 is the intensity of normal conversation (1.0 μW/m² = 1.0 x 10⁻¶ W/m²), D1 is the distance of 30 m, and D2 is the unknown distance. Solving for D2, we get:
D2 = D1 ∙ √(I1/I2)
D2 = 30 m ∙ √(10 W/m² / 1.0 x 10⁻¶ W/m²)
D2 = 30 m ∙ √(1.0 x 10⁷)
D2 = 30 m ∙ 3162.28
D2 = 94868.4 m or approximately 94.9 km
Therefore, to experience the tranquil sound of a normal conversation, which is 1.0 μW/m², you should live at least 94.9 km away from the airport runway.
