Results 1 to 2 of 2

Math Help - Theoretical rate problem

  1. #1
    Newbie
    Joined
    Nov 2006
    From
    Canada
    Posts
    15

    Theoretical rate problem

    I have finished this problem and got some answers, but I don't think there right. If someone can confirm my answers or correct them that would be a big help.

    A woman lives on an island that is 1 km from the mainland. She paddles her canoe at 3 km/h and jogs at 5 km/h. She would like to go to the drug store that is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time?

    There are three parts I did. First I found the time it would take for her to canoe straight across and jog the rest of the way and got 0.56 hours. Next I calculated her canoeing straight to the drug store and got 1.05 hours. The last part would be somewhere inbetween and I got 0.612 hours. If all of that is correct then her landing point should be 52.1 m from the drug store.

    Please if anyone can confirm if my answers are correct or show me your solution with explination if mine is wrong, it would help me out significantly.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by dcfi6052 View Post
    ...

    A woman lives on an island that is 1 km from the mainland. She paddles her canoe at 3 km/h and jogs at 5 km/h. She would like to go to the drug store that is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time?

    ...
    Hello,

    I've attached a rough sketch of the situation.

    speed=\frac{distance}{time}. Thus

    time=\frac{distance}{speed}

    First calculate the distances:

    s=\sqrt{1+x^2} is the distance which is travelled by canoe.

    l = 3 \text{ km} - x is the distance to jog.

    Now use the formula to calculate the time needed:

    t(x)=\frac{\sqrt{1+x^2}}{3}+\frac{3-x}{5}

    This is a function of t in x. Calculate the first derivative:

    t'(x)=\frac{1}{6} \cdot (1+x^2)^{-\frac{1}{2}} \cdot 2x - \frac{1}{5}. To get the minimum t'(x) = 0:

    \frac{x}{\sqrt{1+x^2}}=\frac{3}{5}. Square the equation and multiply by (1+x≤). You'll get \frac{16}{25} x^2=\frac{9}{25}

    The only possible result is x = 0.75

    Plug in this value in t(x) and you get the minimum time: 0.86666 ... = 13/15 hours = 52 minutes.

    EB
    Attached Thumbnails Attached Thumbnails Theoretical rate problem-canu_drugstore.gif  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Theoretical Exercise
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: October 4th 2010, 09:21 PM
  2. Theoretical Stats Problem
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 10th 2010, 03:30 PM
  3. Replies: 1
    Last Post: May 26th 2009, 11:25 PM
  4. Theoretical probability
    Posted in the Statistics Forum
    Replies: 6
    Last Post: August 31st 2008, 04:21 AM
  5. Theoretical yield
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: May 18th 2008, 06:03 AM

Search Tags


/mathhelpforum @mathhelpforum