Solve Applied Problems Involving Hyperbolas
Solve the problem.
-Two LORAN stations are positioned 278 miles apart along a straight shore. A ship records a time difference of 0.00086 seconds between the LORAN signals. (The radio signals travel at 186,000 miles per
Second.)Where will the ship reach shore if it were to follow the hyperbola corresponding to this time
Difference? If the ship is 200 miles offshore, what is the position of the ship?
A)59 miles from the master station, (161.9, 200)
B)80 miles from the master station, (200, 161.9)
C)59 miles from the master station, (200, 161.9)
D)80 miles from the master station, (161.9, 200)

Question

Quizplus · Accepted Answer

The difference in distance from the ship to the two stations is the speed of the signal times the time difference: $186,000 \times 0.00086 = 159.96$ miles, approximately 160 miles. The ship forms a hyperbola with the foci (stations) 278 miles apart. If the ship is 200 miles offshore and the difference in distances to the stations is 160 miles, it's closer to one station than the other. The correct position is 59 miles from the master station, making the coordinates (161.9, 200) when considering the geometry of the situation and the definition of a hyperbola.