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Math Help - Calculus related rates

  1. #1
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    Calculus related rates

    A plane flying horizontally at a altitude of 4mi and a speed of 440mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10mi away from the station.

    Thanks!
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  2. #2
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    Hello, asnxbbyx113!

    A plane flying horizontally at an altitude of 4 miles and a speed of 440 mph
    passes directly over a radar station.
    Find the rate at which the distance from the plane to the station is increasing
    when it is 10 miles away from the station.
    Code:
                     x
          + - - - - - - - - - - - *
          |                    *
          |                 *
        4 |              *
          |           * R
          |        *
          |     *
          |  *
          * - - - - - - - - - - - -

    x is the distance the plane has flown horizontally.
    R is the plane's distance from the station.

    We have: . R^2 \:=\:x^2 + 4^2

    Differentiate with respect to time: . 2R\left(\frac{dR}{dt}\right) \:=\:2x\left(\frac{dx}{dt}\right)

    . . and we have: . \frac{dR}{dt} \:=\:\frac{x}{R}\left(\frac{dx}{dt}\right)


    When R = 10, we have: . 10^2\:=\:x^2 + 16\quad\Rightarrow\quad x \:=\:\sqrt{84}
    . . and we are given: . \frac{dx}{dt} \,=\,440

    Therefore: . \frac{dR}{dt}\:=\:\frac{\sqrt{84}}{10}(440) \:=\:88\sqrt{21} mph.

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