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Math Help - Rectangles Perimeter??

  1. #1
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    Rectangles Perimeter??

    The pattern below is composed of rectangles. This pattern is used repeatedly to completely cover a rectangular region 12L units long and 10L units wide. How many rectangles of dimension L by W are needed?

    Can you please explain why? Thank you in advance.


    - - - - - - -
    550!
    Attached Thumbnails Attached Thumbnails Rectangles Perimeter??-pg-550.jpg  
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  2. #2
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    Didn't we just do this somehwere else?

    You need three things:

    1) Conversion: Looking at the vertical labels in your not-to-scale drawing, one sees that 3W = 2L or W = (2/3)L. This makes the width of your figure L + (2/3)L = (5/3)L.

    2) Reassurance: As luck would have it, in a textbook problem, we have 10L / [(5/3)L] = 6 AND 12L / 2L = 6. This means we CAN tile it without cutting anything. This also suggests how to orient the figure in the final construction.

    3) Direction: 550 is WAY to many. Just adding the area of the pieces, 550*L*W = 550*L*(2/3)L = [366.66...]L^2 total area. Whereas, (10L)*(12L) = 120L^2. With 550, you can do the whole job THREE TIMES and have a little left over.
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  3. #3
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    Hello, fabxx!

    An elaboration of TKHunny's solution . . .


    The pattern below is composed of rectangles.
    The rectangle is used repeatedly to cover a rectangular region 12L \times 10L.
    How many rectangles of dimension L \times W are needed?
    Code:
              W         L
          * - - - * - - - - - *
          |       |           |
          |       |           | W
        L |       |           |
          |       * - - - - - *
          |       |           |
          * - - - *           | W
          |       |           |
          |       * - - - - - *
        L |       |           |
          |       |           | W
          |       |           |
          * - - - * - - - - - *

    As TKHunny pointed out: . 2L = 3W \quad\Rightarrow\quad W = \frac{2}{3}L

    So the width of the rectangle is \frac{5}{3}L . . . its height is 2L.

    Since 6\times \frac{5}{3}L \:=\:10L, we can put 6 rectangles in a row to cover 10L.

    Since 6 \times 2L \:=\:12L, we can make 6 rows to cover 12L.


    Therefore, it takes: 6 \times 6 \:=\:{\color{blue}36} rectangles to cover the region.

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  4. #4
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    36 of the GROUP rectangles, making it 36*5 = 180 rectangles of size L*W.

    As stated previously, 550 >> 180.
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