Triangulation Via Hyperbolas -Epsilon
The long range navigation system, or LORAN, uses hyperbolas to allow ships to determine their exact
location using radio transmi#ers. The objective of this problem was to apply the principles of LORAN and
properties of hyperbolas to determine the exact location of a ship. In the given scenario, three
transmi#ers gave signals at regular intervals traveling 0.3 kilometers per microsecond (speed of light). On
the coordinate plane, Transmi#er #1 was located at (0,0), Transmi#er #2 was located at (0,100), and
transmi#er three was located at (0,400). In order to solve this problem, the initial step was to draw a
graph and label the transmi#ers, the ship, and the distances from the transmi#ers to the ship (see right).
The next step was to specify the distances in terms of the time it took for the signal to reach the ship
from Transmi#er #2 (t microseconds). Although this exact time was not given, we did know that the ship
received the signal from Transmi#er #1 36.5907 microseconds a5er receiving it from Transmi#er #2. It
also received the signal from Transmi#er #3 302.7756 microseconds a5er receiving it from Transmi#er
#2. Thus:
d1 = 0.3km/ms * t
