calculus

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Boat A leaves the dock at 12pm heading due South at a constant speed of 20 km/h.
Meanwhile, at 12pm Boat B, a pirate ship, is 15 km due West of the dock where boat
A started, and is sailing due East at a constant speed of 15 km/h. If the pirates get
within 10 km of another boat, they will spot it and eventually capture it. Is Boat A
in danger of getting captured by the pirates on Boat B?

  • calculus -

    Make a diagram showing the ships at t hours after 12:00 pm.
    Let D be the position of the dock and A and B the position of the boats.
    then DA = 20t km
    and BD = (15-15t) km

    AB^2 = (20t)^2 + (15-15t)^2

    let's find the minimum distance AB
    2AB d(AB)/dt = 800t + 2(15-15t)(-15)
    = 0 for a minimum of AD
    800t - 30(15-15t) = 0
    800t - 450 + 450t =
    t = 450/1250
    t = .36 hours
    when t = .36
    (AB)^2 = 51.84 + 92.16
    AB = √144
    = 12

    Since they have to be within 10 km to be seen, they are ok.

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