Concept explainers
Fireworks Display Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time the first person, standing at point , hears the burst. One second later the second person, standing at point , hears the burst. If the person at point is due west of the person at point , and if the display is known to occur due north of the person at point , where did the fireworks display occur?
To find: Where the fireworks occurred.
Answer to Problem 75AYU
Fire display occurred at due north of the person at .
Explanation of Solution
Given:
Two people are standing 2 miles apart both see the burst from a fireworks display. After a period of time the first person, standing at point , hears the burst. One second later the second person, standing at point , hears the burst. The person at point is due west of the person at point , and the display is known to occur due north of the person at point ,
Formula used:
Equation of the hyperbola , .
Calculation:
Let be location of firework and let it be a point north of such that time difference would be same as that for the first burst would. All such points form a hyperbola with and as foci and on it.
Let contain and let origin be midpoint of .
Sound travels at 1100 feet per second.
So person at is 1100 feet closer to the location than .
Hence difference between to and to is a constant 1100.
Let the equation of the hyperbola be ,
; ,
Distance between and is ; .
To find
Substituting and , we get,
Let ,
Solving the above equation,
We get .
Hence, fire display occurred at due north of the person at .
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