Hyperbola Word Problem: Meteorologists and the Speed of Sound

[js14]

Two amateur meteorologist, living 4km apart (4000m), see a storm approaching. The one farthest from the storm hears a loud clap of thunder 9 sec after the one nearest. Assuming the speed of sound is 340m/sec, determine an equation that models possible locations for the storm at that time.

The graph associated with this problem is a horizontal graph with foci at + or - 2. It has the vertices right between + or - 1 and 2. This is an extra credit problem, and I'm kinda confused on how I should begin this...

 

[HallsofIvy]

The first thing you need to do is set up a coordinate system. It appears that, here, the coordinate system was set up with the x-axis running through the positions of the two people and the origin exactly half way between them. You are wrong to say the foci "are at + or - 2". The foci are points in the plane and require two coordinates- the foci are at (2, 0) and (-2, 0), the positions of the people. Suppose lightning strikes at (x, y). The distance to the one person is $\sqrt{(x-2)^2+ y^2}$. Taking s as the speed of sound, the time it will take that person to hear the clap of thunder is
$$\frac{\sqrt{(x-2)^2+ y^2}}{s}$$
The distance to the other person is $\sqrt{(x+ 2)^2+ y^2}$. The time it will take that person to hear the clap of thunder is
$$\frac{\sqrt{(x+ 2)^2+ y^2}}{s}$$

Set the difference of those equal to 9 and simplify.