Andrew, located at (0, -2200) fired a rifle. The sound echoed off a cliff at (0,2200) to Brian, located at point (x,y). Brian heard the echo 6 seconds after he heard the original shot. Find the xy equation of the curve on which Brian is located. Assume the distances are in feet and that sound travels at 1100 feet/second. (hint: find the equation of the hyperbola)

Remember that a hyperbola is the set of all points for which the difference of distances from two foci (fixed points) is constant. You have the co-ordinates of the foci already, so all you need is the (constant) difference.

To determine this quantity, you can (conveniently) choose any point on the hyperbola you wish. Let's choose the vertex on the y axis, closest to the cliff (that is, the one at (0, +y) ).

Now we do a little physics. Brian hears the original sound at time t = distance / speed = (y + 2200) / 1100. The sound runs right through Brian carrying on to the cliff (2200 - y) feet away, and returns to him in 6 seconds. So delta-t is 6 seconds and it equals 2 ( 2200 - y) / 1100. Re-arrange this to solve for the value y.

With the locations of the foci and now a location of one vertex, you can easily calculate the difference of distances and/or the equation of the hyperbola.

I hope this helps!
Stephen La Rocque>