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Math Help - Hyperbola word problem

  1. #1
    Junior Member Greymalkin's Avatar
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    Hyperbola word problem

    Hyperbola word problem-trig1.pngHyperbola word problem-trig2.png

    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.
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    Senior Member MaxJasper's Avatar
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    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

    Attached Thumbnails Attached Thumbnails Hyperbola word problem-hyperbola.png  
    Last edited by MaxJasper; August 22nd 2012 at 05:45 PM.
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  3. #3
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    Re: Hyperbola word problem

    Quote Originally Posted by Greymalkin View Post
    Click image for larger version. 

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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 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.
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  4. #4
    Junior Member Greymalkin's Avatar
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    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.
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  5. #5
    Senior Member MaxJasper's Avatar
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    Lightbulb 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?
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