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Math Help - AIRPLANE Problem

  1. #1
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    Question AIRPLANE Problem

    Two airplanes (at the same height) are flying away from an airport at a right angle to each other.
    The distance between them is 470 miles.
    Plane X is 270 miles from the airport.
    Plane Y is traveling at a speed of 520 mi/hr.
    The distance between them is increasing by 660 mi/hr.

    How fast is plane X traveling?
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  2. #2
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    Hello, McDiesel!

    Two airplanes are flying away from an airport at a right angle to each other.
    The distance between them is 470 miles.
    Plane X is 270 miles from the airport.
    Plane Y is traveling at a speed of 520 mi/hr.
    The distance between them is increasing by 660 mi/hr.

    How fast is plane X traveling?
    Code:
        Q *
        : |  *
        y |     *
        : |        *
        C *           *   z
        : |  *           *
       ___|     *  470      *
    20√370|        *           *
        : |           *           *
        : |              *           *
        - * - - - - - - - - * - - - - - *
          A       270       B     x     P

    The airport is at A.

    Plane X is at B\!:\;AB = 270

    Plane Y is at C\!:\;BC = 470.
    In right triangle CAB\!:\;\;AC^2 + 270^2 \:=\:470^2 \quad\Rightarrow\quad AC \:=\:20\sqrt{370}


    In a certain time, plane X flew x miles from B to P.
    . . AP \:=\:x+270

    In a certain time, plane Y flew y miles from C to Q\!:\;\;\tfrac{dy}{dt} = 520\text{ mph}
    . . AQ \:=\:y + 20\sqrt{370}

    The distance between them is: . z \:=\:PQ\:\text{ and }\:\tfrac{dz}{dt} = 660\text{ mph}
    From right triangle QAP\!:\;z^2 \;=\;(x+270)^2 + (y + 20\sqrt{370})^2

    Differentiate with respect to time: . 2z\,\frac{dz}{dt} \;=\;2(x+270)\,\frac{dx}{dt} + 2(y + 20\sqrt{370})\,\frac{dy}{dt}

    . . and we have: . (x+270)\,\frac{dx}{dt} \;=\;z\,\frac{dz}{dt} - (y + 20\sqrt{370})\,\frac{dy}{dt}


    At that instant in question: . \begin{Bmatrix}x &=& 0 \\ y&=&0 \\ z &=& 470 \\ \frac{dy}{dt} &=& 520 \\ \\[-4mm] \frac{dz}{dt} &=& 660 \end{Bmatrix}

    Substitute those values and solve for \frac{dx}{dt}

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