Hello, tle8807!

You don't need hyperbolas for this . . . just Pythagorus.

Two people, standing 1 mile apart, both see a flash of lightning.

After a period of time, a person at point 'A' hears thunder.

Two seconds pass, and then person at point 'B' hears thunder.

Assume that person at point 'B' is due West of person at point 'A'

and the lightning strike is known to have occured due North of person at point 'A'.

Where did the lightning strike? Graph the solution.

Sound travels at 1100 ft. per second

1 mile is 5280 ft. Code:

L
*
/ |
/ |
/ |
x+2200 / |
/ | x
/ |
/ |
/ |
* - - - - - - - - *
B 5280 A

$\displaystyle AB = 5280$ feet.

The lightning strikes at $\displaystyle L.$

Let $\displaystyle x = AB$

Point $\displaystyle B$ is 2200 feet further from $\displaystyle L$: .$\displaystyle BL \:=\:x+2200$

Pythagorus says: .$\displaystyle x^2 + 5280^2 \:=\:(x+2200)^2$

. . and we have: .$\displaystyle 4400x \:=\:23,038,400\quad\Rightarrow\quad x \:=\:5236$ feet.