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Math Help - Conic sections problem

  1. #1
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    Conic sections problem

    Fireworks Display Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time, the first person standing at point A hears the burst. One second later, the second person standing at point B hears the burst. If the person at point B is due west of the
    person at point A and if the display is known to occur due north of the person at point A, where did the fireworks
    display occur?

    I tried using an ellipse, but I did not get anywhere. If the speed of sound is v=1129ft/s and the lightning is due north of A, then can't I use the fact that the sum of the distances from the foci is a constant? What am I doing wrong?
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  2. #2
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    Re: Conic sections problem

    I think all you need is the Pythagorean theorem. Am I missing something? Would you not have two intersecting circles?
    Thanks from christianwos
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  3. #3
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    Re: Conic sections problem

    But if you use Pythagoras you get x=2miles, y=vt, z=v(t+1) and Z^2=X^2+Y^2 and when I sub in for x and y I get a negative time.
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  4. #4
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    Re: Conic sections problem

    I'm at a Y on a Smartphone at a disadvantage but l worked this out on Wolfram Alpha & l got a single + root ... get back to you.. Your hypotenuse is BF the distance you want is AF.. You want to use seconds and feet.
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  5. #5
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    Re: Conic sections problem

    l got 43.243 s and.9.24 mile.
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  6. #6
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    Re: Conic sections problem

    I know, you are right. I did not think of converting the distance to feet. I got it, thanks.
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  7. #7
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    Re: Conic sections problem

    Hello, christianwos!

    You didn't convert the two miles to feet.
    (Neither did I, the first time through.)


    Fireworks Display
    Suppose that two people standing 2 miles apart; both see the burst from a fireworks display.
    After a period of time, the first person standing at point A hears the burst.
    One second later, the second person standing at point B hears the burst.
    If point B is due west of point A and if the display occurs due north at point A,
    where did the fireworks display occur?

    We don't need ellipses . . . We have a right triangle.

    Code:
                            F    
                            *    
                         *  |    
            v(t+1)    *     |      
                   *        | vt
                *           |
             *              |
        B * - - - - - - - - * A
                    x
    Let v = 1129\text{ ft/sec}
    Let x = 10,\!560\text{ ft}

    A and B are x feet apart.
    The fireworks are at }F, due north of A.

    In t seconds, the sound travels vt feet from F to A.
    In t+1 seconds, the sound travels v(t+1) feet from F to B.

    Pythagorus: . (vt)^2 + x^2 \:=\:[v(t+1)]^2

    . . . . . . . . . . v^2t^2 + x^2 \:=\:v^2t^2 + 2v^2t + v^2

    . . . . . . . . . . . . . 2v^2t \:=\:x^2-v^2

    . . . . . . . . . . . . . . . t \:=\:\frac{x^2-v^2}{2v^2}


    Hence: . t \:=\:\frac{10,\!560^2 - 1129^2}{2(1129^2)} \:=\:43,24314023 \:\approx\:43.24\text{ sec}

    Then: . F\!A \:=\:vt \:=\:1129(43.24) \:\approx\:48,\!817.96\text{ ft} \:\approx\: 9.25\text{ miles}


    Therefore, the fireworks display is about 9.25 miles north of A.
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