To determine the rate at which the jet airplane radiates energy in the form of sound waves, we can follow these steps:
1. Determine the Surface Area of the Sphere at Initial Distance:
The sound wave from the jet airplane radiates equally in all directions, so we can think of the sound wave spreading out over the surface of a sphere centered on the airplane.
The formula for the surface area [tex]\(A\)[/tex] of a sphere is:
[tex]\[
A = 4 \pi r^2
\][/tex]
where [tex]\(r\)[/tex] is the radius (or distance from the source). Given the initial distance is 3.60 meters, the surface area at this distance is:
[tex]\[
A_{\text{initial}} = 4 \pi (3.60 \, \text{m})^2
\][/tex]
After calculating, the surface area is approximately:
[tex]\[
A_{\text{initial}} \approx 162.86 \, \text{m}^2
\][/tex]
2. Calculate the Power Radiated:
The power radiated by the jet airplane in the form of sound waves can be determined using the intensity of the sound wave and the surface area of the sphere at the initial distance.
The formula for power [tex]\(P\)[/tex] is:
[tex]\[
P = I \cdot A
\][/tex]
where [tex]\(I\)[/tex] is the intensity of the sound wave and [tex]\(A\)[/tex] is the surface area of the sphere. Given the intensity is 101 W/m[tex]\(^2\)[/tex], the power radiated is:
[tex]\[
P = 101 \, \text{W/m}^2 \times 162.86 \, \text{m}^2
\][/tex]
Calculating this gives approximately:
[tex]\[
P \approx 16448.88 \, \text{W}
\][/tex]
3. Convert Power to Kilowatts:
To express the power in kilowatts (kW), we need to convert watts to kilowatts. Since 1 kilowatt (kW) is equal to 1000 watts (W):
[tex]\[
P_{\text{kw}} = \frac{P}{1000}
\][/tex]
Thus,
[tex]\[
P_{\text{kW}} = \frac{16448.88 \, \text{W}}{1000}
\][/tex]
After conversion, the power is approximately:
[tex]\[
P_{\text{kW}} \approx 16.45 \, \text{kW}
\][/tex]
Therefore, the jet airplane radiates energy in the form of sound waves at a rate of approximately [tex]\(16.45 \, \text{kW}\)[/tex].
