Question

Suppose that two people standing 6 miles apart both see the burst from a fireworks display. After a period of​ time, the first person standing at point A hears the burst. Four seconds ​later, the second person standing at point B hears the burst. If the person at point B is due west of the person at point A and if the display is known to occur due north of the person at point​ A, where did the fireworks display​ occur? Note that sound travels at 1100 feet per second.
The fireworks display is_____feet north of the person at point A

Answer 1

Answer:

$x=1824218 ft$

Step-by-step explanation:

The figure attached show a general description of the problem.

For this case the best way to solve the problem is use the general equation for an hyperbola given by:

$\frac{y^2}{a^2} -\frac{x^2}{b^2}=1$

We assume that the longer axis for the hyperbola is on the y axis.

On this case the total distance traveled by the sound is 1100 and we can find the value of a like this:

$2a = 1100$

$a =\frac{1100}{2}=550$

Since both people are apart 6 miles we can find the value of c like this:

$2c =2(6)$

c= 6 miles and we can convert this into ft like this:

$6miles *\frac{5280ft}{1 mile}=31680ft$

And we can find the value of b like this:

$b^2 = c^2 -a^2 = (31680)^2 -(550)^2=1003319900$

And then our equation is given by:

$\frac{y^2}{302500} -\frac{x^2}{1003319900}=1$

And for this case we want to find the value of x when y = 6ft=31680 ft[/tex], and solving for x we got:

$\frac{31680^2}{302500} -\frac{x^2}{1003319900}=1$

$\frac{x^2}{1003319900} = \frac{31680^2}{302500} -1$

$x=1824218 ft$