Calculus

A 23 foot ladder is leaning on a 6 foot fence. The base of the ladder is being pulled away from the fence at the rate of 7 feet/minute. How fast is the top of the ladder approaching the ground when the base is 6 from the fence? [Note: The ladder is protruding over the top of the fence.]

The top of the ladder is approaching the ground at the rate of____________ feet/minute. [Enter a positive number only.]

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  1. Let the ladder make an angle t with the ground.

    Let the distance from the base of the fence = x

    Let the end of the ladder be at height h.

    tan(t) = 6/x
    sin(t) = h/23

    tan^2 = sin^2/cos^2

    36/x^2 = (h/23)^2 / (1 - (h/23)^2)
    flip it upside down:

    x^2/36 = (1-h^2/529)/(h^2/529)
    x^2/36 = 529/h^2 - 1

    2x/36 dx = -2(529)/h^3 dh

    when x = 6
    tan(t) = 6/6 = 1
    so, h = 23/√2

    2(6)/36 * 7 = -2(529)/(23^3/2√2) dh

    84/36 = -1058/4302 dh

    dh = -9.49 ft/s
    or,

    9.49 ft/s downward

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