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A car is traveling north towards an intersection at 60mph at the same time a truck is headed east toward the same intersection at 45mph. Find the rate of change of the distance between the car and truck when the car is 3 miles south of the intersection and the truck is 4 miles west of the intersection. Is this distance increasing or decreasing at that time?

  • FREE BODY DIAGRAM calculus -

    Place the car on the negative y axis and label the distance to the origin y
    Place the truck on the negative x axis and label the distance to the origin x
    label the distance between them d
    d^2 = x^2 + y^2
    2d dd/dt = 2x dx/dt + 2y dy/dt
    dd/dt =(x dx/dt + y dy/dt)/d

    Given: when x = 4 , dx/dt = -45
    when y = 3 , dy/dt = -60
    d = 5 , (did you notice the 3-4-5 right-angled triangle ?)

    dd/dt = (4(-45) + 3(-60)/5 = -72

    At that moment the distance between them is decreasing at 72 mph

    The distance between them is decreasing, since dd/dt is negative.

  • FREE BODY DIAGRAM calculus -


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