Solution

LORAN (long distance radio navigation) for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light ($186{,}000$ miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, $500$ miles apart, are positioned on a rectangular coordinate system at $(-250, 0)$ and $(250, 0)$ and that a ship is traveling on a path with coordinates $(x, 120)$ (a horizontal line $120$ miles above the $x$-axis). Find the $x$-coordinate of the position of the ship when the time difference between the pulses from the transmitting stations is $1200$ microseconds ($0.0008$ second).