Results 1 to 4 of 4

Math Help - Optimization problem

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    114
    Thanks
    1

    Optimization problem

    Two towns A and B are 5km and 7km, respectively, from a railroad line. The points C and D nearest to A and B on the line are 6 km apart. Where should a station be located to minimize the length of a new road from A to S to B?

    So I set up the question in order to minimize the length of AS^2+SB^2 (since we are not suppose to use trig). I squared the sides in order to make the problem easier to work with. I made x be the length from C to S, and 6-x the length from S to D.

    I have,

    (5^2 + x^2) + (7^2 + (6-x)^2) = 110 - 12x + 2x^2

    differentiating, I get,

    -12+4x

    When I solve for x, I get 3. However, the answer in the back of the book is 21/6=3.5 from point D.

    What am I doing wrong?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,829
    Thanks
    123
    Quote Originally Posted by BrownianMan View Post
    Two towns A and B are 5km and 7km, respectively, from a railroad line. The points C and D nearest to A and B on the line are 6 km apart. Where should a station be located to minimize the length of a new road from A to S to B?

    So I set up the question in order to minimize the length of AS^2+SB^2 (since we are not suppose to use trig). I squared the sides in order to make the problem easier to work with. I made x be the length from C to S, and 6-x the length from S to D.

    I have,

    (5^2 + x^2) + (7^2 + (6-x)^2) = 110 - 12x + 2x^2

    differentiating, I get,

    -12+4x

    When I solve for x, I get 3. However, the answer in the back of the book is 21/6=3.5 from point D.

    What am I doing wrong?
    Did you make a sketch?

    1. The triangles \Delta ASC and \Delta B'SD are similar.

    2. Use the proportion

    \frac x{6-x} = \frac75 Solve for x.
    Attached Thumbnails Attached Thumbnails Optimization problem-orte_anbahn.png  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    Quote Originally Posted by BrownianMan View Post
    Two towns A and B are 5km and 7km, respectively, from a railroad line. The points C and D nearest to A and B on the line are 6 km apart. Where should a station be located to minimize the length of a new road from A to S to B?

    So I set up the question in order to minimize the length of AS^2+SB^2 (since we are not suppose to use trig). I squared the sides in order to make the problem easier to work with. I made x be the length from C to S, and 6-x the length from S to D.

    I have,

    (5^2 + x^2) + (7^2 + (6-x)^2) = 110 - 12x + 2x^2

    differentiating, I get,

    -12+4x

    When I solve for x, I get 3. However, the answer in the back of the book is 21/6=3.5 from point D.

    What am I doing wrong?
    since d = \sqrt{5^2+x^2} + \sqrt{7^2 + (6-x)^2}

    you cannot square "everything" ...

    d^2 \ne (5^2+x^2) + [7^2 + (6-x)^2]
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    The shortest distance between 2 points is a straight line.
    As illustrated by Earboth, in this case,
    the straight line goes to the image of one of the points.
    Attached Thumbnails Attached Thumbnails Optimization problem-road.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Optimization problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 10th 2009, 03:53 AM
  2. Optimization Problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 9th 2009, 05:32 PM
  3. Optimization problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 7th 2008, 05:52 PM
  4. Optimization Problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 16th 2008, 12:21 PM
  5. Optimization Problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 9th 2008, 04:42 PM

Search Tags


/mathhelpforum @mathhelpforum