Jane is 2 miles offshore in a boat and wishes to reach a coastal village 6 miles down a straight shoreline from the point nearest the boat.

Question

Jane is 2 miles offshore in a boat and wishes to reach a coastal village 6 miles down a straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time

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Ngọc Diệp 5 years 2021-08-10T00:12:39+00:00 1 Answers 235 views 0

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  1. hongcuc2
    0
    2021-08-10T00:14:38+00:00
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    Answer:

    x  = 0,87 miles

    Step-by-step explanation:

    Let´s look at the stuation.

    Jane is two miles away (in point B) from point A at the beach  and the horizontal distance between Jane and the village( Point C ) is 6 miles, she will leave the boat somewhere (Point D,  at x distance from A) between A and C.

    Then distances  for the travel of Jane are:

    L₁  distance to row      L₁²  =  2² + x²     L₁  = √ 4 + x²

    And the distance to walk    L₂  =  6 – x

    The relation between distances, time, and speed is

    d = v*t   then  t =  d/v

    According to that the time for rowing will be  

    t₁  = ( √4  +  x² ) / 2      and for walking   t₂  = (  6  –  x  ) /5

    T = t₁  +   t₂

    T(x)  = ( √4  +  x² ) / 2   + (  6  –  x  ) /5

    Tacking derivatives on both sides of the equation

    T´(x)  =  1/2  [ ( 2*x / 2√ ( 4 + x² ) ]  -1/5

    T´(x)  = ( 1/2 )*  x / √ ( 4 + x² )   – 1/5

    T´(x) = 0

    ( 1/2 )*  x / √ ( 4 + x² )   – 1/5   = 0

    5*x  – 2* √ ( 4 + x² )  =  0

    5*x  =  2 *√ ( 4 + x² )

    Squaring

    25*x²  =  4* ( 4 + x² )

    25*x²  =  16  +  4*x²

    21*x²  =  16

    x  =  √ 16/21

    x  = 0,87 miles

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