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):
29°c
1221 m
3735 ms

Explanation:

Step1: Calculate speed of sound

The speed of sound $v$ in air can be approximated by the formula $v = 331 + 0.6T$, where $T$ is the temperature in Celsius. Given $T = 29^{\circ}C$, then $v=331 + 0.6\times29=331+17.4 = 348.4$ m/s.

Step2: Calculate total distance

The time $t = 3735$ ms $=3.735$ s. Using the formula $d=v\times t$, with $v = 348.4$ m/s and $t = 3.735$ s, we get $d=348.4\times3.735 = 1301.274$ m.

Step3: Calculate height of explosion

Assuming the horizontal - distance from the person to the point directly below the explosion is $x = 1221$ m and the slant - distance from the person to the explosion is $d = 1301.274$ m. Using the Pythagorean theorem $h=\sqrt{d^{2}-x^{2}}$, where $d = 1301.274$ m and $x = 1221$ m. First, $d^{2}=(1301.274)^{2}=1693317.57$, $x^{2}=(1221)^{2}=1490841$. Then $h=\sqrt{1693317.57 - 1490841}=\sqrt{202476.57}\approx449.97\approx450$ m.

Answer:

Speed of Sound (m/s): 348.4
Total Distance (m): 1301.274
Height of Explosion (m): 450