I've got two LORAN stations A and B that are 500 miles apart. A and B are also the Foci of a hyperbola. A ship at point P (which lies on the hyperbola branch with A as the focus) receives a nav signal from station A 2640 micro-sec before it receives from B. If the signal travels 980 ft/microsecond, how far away is P from A and B? Also, what are the values for a, b, and c?
Assuming the Transverse axis is horizontal and the center of the hyperbole is the origin, the foci are:
A(250, 0) and B(-250, 0) ; thus c = 250
Now, let's figure out how far appart is P from A and B. This is the fun part. Ready? OK. Since the speed of the signal is given in feet/microsecond (ft/μs), we need to use the unit conversion 1 mile = 5,280 feet. (μs is the abreviation for microsecond... Fancy, huh?)
d = (2,640 μs)(980 ft/μs)
The signal travels 2,587,200 feet; or 490 miles in 2,640 μs
Now, let's think about this. If the stations are 500 miles appart, and the ship receives the signal 2,640 μs sooner from A than from B, it means that the ship is very close to A because the signal traveled 490 additional miles from B before it reached the ship. Let's put the ship P at the vertex of branch A and the vertices are 490 miles appart; or 245 miles from the origin...
Then... a = 245 and the vertices are (245, 0) and (-245, 0)
We find b from the fact: c2 = a2 + b2 → b2 = c2 - a2; or b2 = 2,475; thus b ≈ 49.75
Now the answers to the questions:
The distance from P to A is 5 miles → PA = 5; from P to B is 495 miles → PB = 495.
a = 245; b ≈ 49.75; and c = 250
Let's take a minute to check our facts:
AP = 5 miles or 26,400 ft ÷ 980 μs/ft = 26.94 μs
BP = 495 miles or 2,613,600 ft ÷ 980 μs/ft = 2,666.94 μs
The difference 2,666.94 - 26.94 = 2,640 μs, is exactly the time P received the signal sooner from A than from B.
This was too much fun for a Thursday night. Cheer up, tomorrow is Friday, finally!