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calculus

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Optimization

At 1:00 PM ship A is 30 miles due south of ship B and is sailing north at a rate of 15mph. If ship B is sailing due west at a rate of 10mph, at what time will the distance between the two ships be minimal? will the come within 18 miles of each other?


The answer is 2:23:05 and yes,
i just need help with the setup

  • calculus - ,

    The positions of A and B are parametric functions of time.

    Take the position of ship A at 1:00 pm be (0,0), the
    xa(t) = 0
    ya(t) = 10t

    xb(t) = -15t
    yb(t) = 30

    Distance between the two ships
    D(t) = √((xb(t)-xa(t))²+(yb(t)-ya(t))²)

    Substitute xa,xb,ya,yb in the expression for D(t) and differentiate with respect to t.

    For the minimum distance, D'(t0) = 0.
    Verify that D(t0) is a minimum by confirming that D"(t0)>0, or by comparing values of D'(t0-) and D'(t0+) with D'(t0).

    Evaluate D(t0) for the remainder of the question.

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