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Question 1169029: Two control towers are located at points Q(-400,0) and R(400,0), on a straight shore
where the x-axis runs through (all distances are in meters). At the same moment, both
towers sent a radio signal to a ship out at sea, each traveling at 250 m/µs. The ship
received the signal from Q 2.9 µs (microseconds) before the message from R. Find the
equation of the curve containing the possible location of the ship.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! Ok, I'll give this one a try. I was reminded of the definition of the hyperbola
when the problem mentioned a fixed time difference between the signals arriving
from Q vs. the signals from R. The hyperbola is the set of all points
(x,y) such that the difference of the distances from (x,y) to the foci is constant. If we
place a tower at each focus, then we can satisfy the conditions
of the problem with a hyperbola to describe the possible locations of the ship.
The difference in the distances between a point on the curve and the
two foci = 2a, for the vertex at (a,0). The foci are at the points (-c,0) and (c,0), and b = sqrt(c^2 - a^2).
Since the signal velocity is 250 m/us, the difference in the travel times
from each of the towers is
d2/250 - d1/250 -> d2 - d1 = 250 m/us * 2.9 us = 2a = 725.
Thus d2 - d1 = 250*2.9 = 725 m. Thus a = 725/2 = 362.5
Therefore b = sqrt(400^2 - 362.5^2) = 169.09687
The equation for a hyperbola centered at the origin is (x/a)^2 - (y/b)^2 = 1.
So the possible locations for the ship are given by:
(x/362.5)^2 - (y/169.09687)^2 = 1.
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