Results 1 to 4 of 4

Math Help - Mosaics

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    68

    Mosaics

    Janet and Tom are making a mosaic out of 10cm x 10cm tiles to hang in their living room. The mosaic is going to be 40 tiles high by 60 tiles long and they want to have a border that leaves 84% of the original area for a more detailed design in the centre. How wide should the border be and will it work nicely with tiles of this size?

    I have the answer, i just need to know how to do it

    thankkks : )

    Answer: 20 cm + yes...

    but whyy??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Quote Originally Posted by foreverbrokenpromises View Post
    Janet and Tom are making a mosaic out of 10cm x 10cm tiles to hang in their living room. The mosaic is going to be 40 tiles high by 60 tiles long and they want to have a border that leaves 84% of the original area for a more detailed design in the centre. How wide should the border be and will it work nicely with tiles of this size?

    I have the answer, i just need to know how to do it

    thankkks : )

    Answer: 20 cm + yes...

    but whyy??
    Here's my method, there's probably a neater way than this but oh well

    You mosaic is 40 tiles by 60, this means a total of 2400 tiles.

    If you wish to have 84% left in the middle, this means that the border will 16% of the total mosaic, or 384 tiles.

    Now consider the perimeter of the border. You have two sides of 60, and two sides of 38 (sides of 40, but you will have already counted the two corners with the 60 if that makes sense), this gives you 196 tiles in your first border.

    Now your second border, there are two sides of 58, and two sides of 36 (for the same reason above it's not sides of 38), giving you 188 tiles in your second border.

    Adding these together you get exactly 384, leaving the 84% free in the middle.

    Your border is 2 tiles wide, hence the 20cm.

    Hope this helps
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2009
    Posts
    68
    thankks : )
    i understand it more now.
    but how would you solve this problem using a quadratic equation?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,718
    Thanks
    634
    Hello, foreverbrokenpromises

    The size of the tiles in irrelevant.
    We are concerned with the number of tiles (whole tiles).


    Janet and Tom are making a mosaic to hang in their living room.
    The mosaic is going to be 40 tiles high by 60 tiles long
    and they want to have a border that leaves 84% of the original area in the centre.
    How wide should the border be and will it work nicely with tiles of this size?
    Code:
          : - - -  60 - - - - :
        - *-------------------* -
        : |                   | x
        : |   *---------- *   | -
        : |   |///////////|   | :
       40 |   |///////////|   | 40-2x
        : |   |///////////|   | :
        : |   *-----------*   | -
        : |                   | x
        - *-------------------* -
          : x : - 60-2x - : x :

    The height of the mosaic is 40 tiles, its length is 60 tiles.
    The border will be x tiles wide.

    The height of the shaded region is: . 40-2x tiles.
    The length of the shaded region is: . 60-2x tiles.

    The area of the shaded region is: . (40-2x)(60-2x) \:=\:2400 - 200x + 4x^2 tiles
    . . which will be 84% of (40 60) = 2016 tiles.

    There is our equation! . . . . 2400 - 200x + 4x^2\:=\:2016 \quad\Rightarrow\quad x^2 - 50x + 96 \:=\:0

    . . which factors: . (x - 2)(x - 48) \:=\:0

    . . and has roots: . x \:=\:2,\;\;{\color{red}\rlap{///////}}x\:=\:48

    Therefore, the border will be 2 tiles wide.

    [Since the tiles are 10 cm 10 cm, the border will be 20 cm wide.]

    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum