# Hyperbola word problem

• Aug 22nd 2012, 04:03 PM
Greymalkin
Hyperbola word problem
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Problem above. I see it's based on the LORAN long range navigation system, but my book was very nondescript about it, except that the boat is located where the branches of 2 hyperbolas meet(see 2nd pic). With the given information I am hard pressed to find any equation(I thought there were 2 equations!) because the boat is not equidistant from the transmitters. Trying to find such distances with the difference in signal transmission gives me s=x+37.2 and t=x, but thats about as far as I get without running out of numbers.
• Aug 22nd 2012, 05:13 PM
MaxJasper
Re: Hyperbola word problem
Take ST line as x-axis or major axis of hyperbola.
Midpoint ST is hyperbola's center C.
CS=CT=150 miles = c to focus S or T.
Length of major axis =2a=37.2 miles.
Length of minor axis/2= b = sqrt(c^2-a^2)=sqrt(150^2-(37.2/2)^2) = 148.842 miles.

Equation of hyperbola with center at C: ((x-x0)/a)^2 -((y-y0)/b)^2 = 1.
(x0,y0)=(0,0)= Point C

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• Aug 23rd 2012, 06:13 AM
HallsofIvy
Re: Hyperbola word problem
Quote:

Originally Posted by Greymalkin
Attachment 24577Attachment 24578

Problem above. I see it's based on the LORAN long range navigation system, but my book was very nondescript about it, except that the boat is located where the branches of 2 hyperbolas meet(see 2nd pic). With the given information I am hard pressed to find any equation(I thought there were 2 equations!) because the boat is not equidistant from the transmitters. Trying to find such distances with the difference in signal transmisseon gives me s=x+37.2 and t=x, but thats about as far as I get without running out of numbers.

You seem to be completely misunderstanding the problem. There are NOT two equation and you are NOT to find a specific point. With two Loran stations like that, the points at which there is a specific difference in timing is one hyperbola and the boat can be at any point on a branch of that hyperbola.
• Aug 24th 2012, 04:23 PM
Greymalkin
Re: Hyperbola word problem
If the boat is going parallel from the shoreline, wouldn't the equation be constantly changing to calculate its path? I don't see the point in using a hyperbola, if thats the case it seems more like curve fitting.
• Aug 24th 2012, 04:48 PM
MaxJasper
Re: Hyperbola word problem
Your assumption that the boat is going parallel to the shoreline is not what the main problem defines...the boat is only located 200 miles from shoreline at time=t...there is no indication which direction is moving or with what speed...so the problem is only for a specific time=t=constant.

The problem definition itself specifies hyperbola with foci at S & T where the boat is located...this fixes the hyperbola as solved in my post...the only use of 200 miles from the shore line is to draw a line parallel to x at point y=-200 to intersect the eastern branch of the hyperbola...that is the present location of the boat.

A more realistic problem definition should make it a moving boat in real time with direction, speed, velocity,...and finding its pass ...or probability of the boat being at location=(x,y,t)...etc. Do you have such a problem definition?