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Math Help - Rectangle Dimensions

  1. #1
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    Rectangle Dimensions

    What are the dimensions of a rectangle of a fixed perimeter P that result in the largest area?

    NOTE: What exactly is a fixed perimeter?
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  2. #2
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    Quote Originally Posted by symmetry View Post
    What are the dimensions of a rectangle of a fixed perimeter P that result in the largest area?

    NOTE: What exactly is a fixed perimeter?
    The dimensions of a rectangle of a fixed perimeter, P, that results in the largest area is a square.

    Thus, the dimensions will be P/4 by P/4.

    And thus the largest area will be (P/4)^2.
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  3. #3
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    ok

    Okay...but where did you get p/4?
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  4. #4
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    Quote Originally Posted by symmetry View Post
    Okay...but where did you get p/4?
    Every side of a square is equal, and therefore the perimeter of a square is:

    x + x + x + x, where x is the length of a side,

    Or similarly, 4x = P, where P is the perimeter

    P/4 = x, which is a side of the square

    Thus, the dimensions that result in the largest area has to be P/4 by P/4.
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  5. #5
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    Quote Originally Posted by symmetry View Post
    ...
    NOTE: What exactly is a fixed perimeter?
    Hello,

    normally the perimeter of a rectangle is calculated by:

    p=2 \cdot length+2 \cdot width

    A fixed perimeter means that the result of this sum is a constant. Length and/or width may change but the perimeter has allways the same value.

    EB
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  6. #6
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    Quote Originally Posted by symmetry View Post
    Okay...but where did you get p/4?
    Hello,

    to demonstrate what a fixed perimeter means I've attached a digram which shows rectangles which have all the perimeter 16 cm. As you may see the size of those rectangles differs a lot. You'll find the rectangle with the greatest size in the middle of the triangle. I've drawn this square with some thicker perimeter.

    Hope this helps a little bit further.

    EB
    Attached Thumbnails Attached Thumbnails Rectangle Dimensions-gl_umfg_rechteck.gif  
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  7. #7
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    ok

    I thank you for the notes.

    Earboth:

    You have outdone yourself here. All your replies with diagrams are the best I've seen.
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