Solve Applied Problems Involving Hyperbolas
-Two LORAN stations are positioned 208 miles apart along a straight shore. A ship records a time difference of $0.00097$ 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 150 miles offshore, what is the position of the ship?
A) 14 miles from the master station, $( 274.2,150 )$
B) 90 miles from the master station, $( 150,274.2 )$
C) 14 miles from the master station, $( 150,274.2 )$
D) 90 miles from the master station, $( 274.2,150 )$

Question

Quizplus · Accepted Answer

The difference in distance from the ship to the two stations is given by the time difference multiplied by the speed of the signals, which is $0.00097 \times 186,000 = 180.42$ miles. Given the geometry of the situation and the properties of hyperbolas, the ship's position relative to the midpoint between the stations can be determined. If the ship is 150 miles offshore and the stations are 208 miles apart, the ship's position along the shore (x-coordinate) can be found by solving the hyperbola equation. The correct answer involves calculating the specific position, which results in the ship being 14 miles from the master station, with coordinates $(274.2, 150)$, indicating it is 274.2 miles from one station along the shore and 150 miles offshore.

Solve Applied Problems Involving Hyperbolas -Two LORAN Stations Are Positioned 0.000970.000970.00097

Solve Applied Problems Involving Hyperbolas
-Two LORAN Stations Are Positioned $0.00097$