Chapter 6.4, Problem 53E

Navigation 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 300 miles apart are positioned on a rectangular coordinate system with coordinates $\left(-150,0\right)$ and $\left(150,0\right)$ and that a ship is traveling on a hyperbolic path with coordinates $\left(x,75\right)$ (see figure).

(a) Find the x-coordinate of the position of the ship when the time difference between the pulses from the transmitting stations is 1000 microseconds (0.001 second).

(b) Determine the distance between the port and station A.

(c) Find a linear equation that approximates the ship’s path as it travels far away from the shore.