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Thread: 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 $\displaystyle X$ is 270 miles from the airport.
    Plane $\displaystyle Y$ is traveling at a speed of 520 mi/hr.
    The distance between them is increasing by 660 mi/hr.

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

    The airport is at $\displaystyle A.$

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

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


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

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

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

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

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


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

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

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