calculus

A man starts walking north at 2ft/s from a point P. Twenty-five seconds later a woman starts walking south at 3ft/s from a point 400ft due east of P. At what rate are the people moving apart 50 seconds after the woman starts walking? Do not include units in your answer.

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  1. Man starts north at t = 0 from origin
    Woman starts south at t = 25 from (400,0)
    where are they at t = 75?
    man 2*75 = 150 so at (0,150)
    woman 3*50 = 150 so at (400,-150)
    distance apart is hypotenuse of triangle with legs 300 and 400 so 500 ft
    How fast is this changing?
    The north south difference is changing at 2+3 = 5 ft/s
    The East west distance is not changing at all
    start new t now at 0
    N-S distance = 300 + 5 t
    E-W distance = 400
    h is hypotenuse, the distance
    h^2 = 400^2 + (300+5t)^2
    2 h dh = 0 + 2 (300+5t)(5)dt
    h dh = 5(300+5t) dt
    at h = 500 and t = 0 this is
    500 dh = 5(300) dt
    dh/dt = .01(300) = 3

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  2. let the times passed since the woman started walking be t seconds
    make a diagram
    distance covered by man = 50 + 2t
    distance covered by woman = 3t
    let the distance between them be D
    I see a right-angled triangle with hypotenuse D,
    the vertical line as (50+2t+3t)
    and the horizontal as 400

    D^2 = (5t+50)^2 + 400^2
    2D dD/dt = 2(5t+50)(5) + 0
    dD/dt = 5(5t+50)/D

    when t = 50,
    D^2 = 300^2+400^2
    D = 500

    dD/dt = 5(300)500 = 3

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  3. thank you very much

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