Calculus Maxima/Minima

An automobile traveling at the rate of 30 ft/sec is approaching an intersection. When the automobile is 120 ft. from the intersection, a truck, traveling at the rate of 40 ft/sec on a road at right angles with the automobile’s road crosses the intersection. How long after the truck leaves the intersection the two vehicles are closest?

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  1. Let the time passed since the truck went through the intersection be t seconds
    Make a diagram
    Let the distance between them be D ft
    I see that

    D^2 = (40t)^2 + (120-30t)^2
    = 1600t^2 + 14400 - 720t + 900t^2
    = 2500t^2 - 7200t + 14400

    2D dD/dt = 5000t -7200
    but at a minimum of D , dD/dt = 0 , so

    5000t = 7200
    t = 7200/5000
    = 1.44 sec after the truck entered the intersection

    check:
    when t = 0 , D = 120
    when t = 1 , D = 98.5
    when t = 1.44 , D = 96
    when t = 1.5 , D = 96.05 , getting bigger again

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  2. thx reiny

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  3. thanks mike

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