Question

a firework explodes high in the sky. you know you are 400m away from the explosion and you measure the sound intensity at your location to be 0.01W/m^2, how much sound power does the explosion emit

Answer 1

The sound power emitted by the explosion can be calculated using the formula for sound intensity, which is 20,118.72 watts

Step-by-step explanation:

Given:

Distance from explosion (r) = 400 meters

Sound intensity (I) = 0.01 W/m²

Calculating Area (A) of the Sphere:

The formula for the area of a sphere in 3D space is

A = 4 * π * r²

where r is the radius (distance from the source).

Given the distance from the explosion (r) is 400 meters:

A = 4 * π * (400)²

A = 4 * π * 160,000

A ≈ 2,011,872.44 m²

Determining Sound Power (P) using Sound Intensity Formula:

Sound intensity (I) = Sound power (P) / Area (A)

Rearrange the formula to solve for sound power (P):

P = I * A

Given the sound intensity (I) is 0.01 W/m² and the calculated area (A) is approximately 2,011,872.44 m²:

P = 0.01 W/m² * 2,011,872.44 m²

P ≈ 20,118.72 watts

In summary, the sound power emitted by the explosion is 20,118.72 watts, calculated using the known sound intensity at the given distance and applying the formula for sound intensity, taking into account the area over which the sound spreads.

Answer 2

The sound power emitted by the firework explosion is 20,096 watts, calculated by multiplying the measured sound intensity at the location (0.01 W/m^2) by the area of the sphere with a radius equal to the distance from the explosion (400 meters).

Step-by-step explanation:

The sound power of a firework explosion can be determined by using the sound intensity at a known distance. The intensity (I) of sound at a distance (r) from a point source that spreads sound uniformly in all directions is given by the formula I = P / (4πr2), where P is the sound power in watts and r is the distance from the source in meters. In this case, we can rearrange the formula to solve for P:

P = I × (4πr2).

Given that the sound intensity at the student's location is 0.01 W/m2 and the distance to the explosion is 400 meters, we can calculate the total sound power of the explosion as follows:

P = 0.01 W/m2 × (4π × (400 m)2)

P = 0.01 W/m2 × (4π × 160000 m2)

P = 0.01 W/m2 × 2009600 m2

P = 20,096 watts

Therefore, the sound power emitted by the firework explosion is 20,096 watts.