Ben asked in Science & MathematicsMathematics · 1 decade ago

Calculus problem help?

Bob is at point B 25 miles from A. Alice is in a boat in the sea at point C 1.75 miles from the beach. Alice rows at 2.5 miles per hour and walks at 4 miles per hour , where along the beach should she land so that she may get to Bob in the least amount of time ?

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  • M3
    Lv 7
    1 decade ago
    Favorite Answer

    .. C

    .. |\ angle ACP = z.

    .. |-\

    w|--\

    ...|----\

    .. |-----\------

    ..A. x P ...B

    ...<---- l --->

    let speed ratio land:water = k (4/2.5)

    dist. over water = w•sec z,

    dist. over land = l - w•tan z

    time taken ∞ k•sec z +(l - tan z)

    dT/dz = k•secz•tan z - sec^2 z

    = (ksin z - 1) / cos^2 z

    for minima, sin z = 1/k

    which yields x = w /√(k^2 - 1)

    plugging in,

    x = 1.75/√(1.6^2 - 1)

    ans: 1.401 km to 3 dp [ calculation error rectified ]

    ------

    note that the optimal value of angle z is solely dependent on k !

  • ted s
    Lv 7
    1 decade ago

    DRAW the figure...let x = distance along the beach where boat beaches , tb = boat time and tw = walk time....time = tb + tw = [ √(1.75² + x²) / 2.5 ] + [ (25 - x ) / 4 ]...find x which minimizes time...you do the computations

    { (7/4) / √15 }....lot of walking....sure that Bob is 25 miles away ??

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