Question

height of firework
use the temperature of the air to find the speed of sound and use the time for the sound to arrive to find the total distance from person to explosion
finally, find the height of the firework when it exploded realizing that the picture below is not to scale
enter answers dont include units
speed of sound (m/s):
total distance (m):
height of explosion (m):
3681 ms

Explanation:

Step1: Find speed - of - sound formula

The speed of sound $v$ in air can be approximated by the formula $v = 331+ 0.6T$, where $T$ is the temperature of the air in degrees Celsius. Since the temperature is not given here, if we assume standard temperature $T = 20^{\circ}C$, then $v=331 + 0.6\times20=331 + 12=343$ m/s.

Step2: Calculate total distance

We know that distance $d=v\times t$. Given $t = 3681$ ms $=3.681$ s and $v = 343$ m/s, then $d=343\times3.681 = 343\times(3 + 0.6+0.08 + 0.001)=343\times3+343\times0.6+343\times0.08+343\times0.001=1029+205.8+27.44+0.343 = 1262.583$ m.

Step3: Assume horizontal distance is zero (simplified case)

If we assume the horizontal distance from the person to the point directly below the explosion is zero (simplified for finding height), the height of the explosion $h$ is equal to the total distance $d$. So $h = 1262.583$ m.

Answer:

Speed of Sound (m/s): 343
Total Distance (m): 1262.583
Height of Explosion (m): 1262.583