Results 1 to 4 of 4

Math Help - Minimization of the distance between 3 points

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    11

    Minimization of the distance between 3 points

    Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation.

    Centerville is located at in the -plane, Springfield is at , and Shelbyville is at . The cable runs from Centerville to some point on the -axis where it splits into two branches going to Springfield and Shelbyville. Find the location that will minimize the amount of cable between the 3 towns and compute the amount of cable needed.


    Not sure how to start on this one. I know that the distance formula is needed, I'm just not sure how.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    May 2010
    Posts
    5
    This looks like an optimalization problem. What you need to do is make a function T(x) for the amount of cable that is needed depending on the "branching distance" x from the origin. Then you need to differentiate that formula, and equal it to 0 to find the extremum of the function. (In this case you'll want a minimum, since you're looking for the least possible amount of cable needed).

    In this case, the formula for the amount of cable needed would be

    T = 11-x + 2\sqrt{25+x^2}

    In which 11-x is the amount of cable from Centerville to the branching point, and the other term is the amount of cable from the two branches (using Pythagoras).
    Last edited by AbAeterno; May 29th 2010 at 09:49 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by momopeaches View Post
    Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation.

    Centerville is located at in the -plane, Springfield is at , and Shelbyville is at . The cable runs from Centerville to some point on the -axis where it splits into two branches going to Springfield and Shelbyville. Find the location that will minimize the amount of cable between the 3 towns and compute the amount of cable needed.


    Not sure how to start on this one. I know that the distance formula is needed, I'm just not sure how.
    Take x as being the distance from the y-axis to the connecting point of the cables.
    Then the distance from x to Centerville is 11-x.

    The distance formula is Pythagoras' theorem in co-ordinate form.

    The arms are governed by Pythagoras' theorem

    x^2+5^2=L^2

    L=\sqrt{5^2+x^2}=\left(5^2+x^2\right)^{0.5}

    On the graph, the tangent will be horizontal at a minimum, hence

    \frac{d}{dx}\left[11-x+2\left(5^2+x^2\right)^{0.5}\right]=0

    -1+2(0.5)\left(5^2+x^2\right)^{-0.5}(2x)=0

    Be careful that you don't state that the horizontal length plus one arm must be a minimum, as you'd have a straight line from Springfield to Centerville.

    \frac{2x}{\sqrt{5^2+x^2}}=1

    2x=\sqrt{5^2+x^2}

    4x^2=25+x^2

    3x^2=25

    x=\sqrt{\frac{25}{3}}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    May 2010
    Posts
    11
    Thanks to both of you! I just wasn't sure how to set the problem up. Seems easy now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. distance between 2 points
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 23rd 2010, 11:55 AM
  2. Distance Between Two Points
    Posted in the Pre-Calculus Forum
    Replies: 10
    Last Post: February 3rd 2010, 10:59 AM
  3. distance Between Two Points
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: March 1st 2009, 08:01 PM
  4. Distance between points
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 24th 2008, 05:14 PM
  5. Distance between points
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: June 6th 2006, 12:06 PM

Search Tags


/mathhelpforum @mathhelpforum