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calculus

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A ladder 25 m long leans against a vertical all. If the lower end is being moved away from the wall at the rate 6m/sec, how fast is the height of the top decreasing when the lower end is 7m from the wall?

  • calculus - ,

    at the moment in question, the ladder top is 24m up the wall.

    x^2+y^2 = 25^2
    2x dx/dt + 2y dy/dt = 0
    2(7)(6) + 2(24) dy/dt = 0
    dy/dt = -4/7 m/s

  • calculus - ,

    start by drawing a diagram. you should notice that you have a triangle with the ladder as the hypoteneuse. you are given that the lower end is moving away from the wall at 6m/sec, so your dx/dt =6.
    To find y when x=7:
    x^2 +y^2 = 25^2
    7^2 +y^2 = 625
    49 + y^2 = 625
    y^2 = 576
    y = 24
    Then, since we know that x is changing:
    x^2 + y^2 = 25^2
    2x(dx/dt) + 2y(dy/dt) = 0 divide by 2:
    x(dx/dt) + y(dy/dt) = 0
    plug in values that you know:
    7(6) + 24(dy/dt) = 0
    24(dy/dt) = -42
    dy/dt = -1.75 m/sec

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