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Math Help - word problem minimum distance

  1. #1
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    word problem minimum distance

    this question is from my grade 12 calculus class I tried working it out but am having trouble drawing a correct diagram, which is the most important step for solving.

    A power boat travels west at 24 km/h. At the instant it passes a buoy, a sailboat sailing north at 7 km/h is 25 km south of the buoy. Calculate the positions of the vessels when there is a minimum distance between them.

    It would help if someone could help with the diagram to help get me started.
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  2. #2
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    Quote Originally Posted by math619 View Post
    ...
    A power boat travels west at 24 km/h. At the instant it passes a buoy, a sailboat sailing north at 7 km/h is 25 km south of the buoy. Calculate the positions of the vessels when there is a minimum distance between them.
    ...
    Hello,

    I've attached a diagram as you requested.

    Let d be the distance between the 2 boats.
    Let t be the time the boats traveled since the powerboat passed the buoy.

    Use Pythagorian rule:

    d^2=(24t)^2+(25-7t)^2. Expand the brackets and collect equal powers:

    d^2=625t^2-350t+625. d² has a minimum if d has a minimum. Thus it is sufficient if you calculate the minimum of d² = D

    D'(t)=1250t - 350. D'(t) = 0 if t=\frac{350}{1250}=\frac{7}{25}

    That means after 7/25 hours = 16 min 48 s, the distance is a minimum. Plugin this value for t and calculate d.

    d = 24 km

    EB
    Attached Thumbnails Attached Thumbnails word problem minimum distance-sail_power.gif  
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