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12 June, 12:08

What is the length of a rectangle if the area is 24 meters squar and the height of 4 meters

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  1. 12 June, 14:15
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    We are given a rectangular yard with area of 24 square meters, and we are asked to find the possible perimeters.

    Since the yard is a rectangle (a parallelogram with right angles), we can describe the dimensions as length and width. The area is given by A=lw and the perimeter by P=2l+2w.

    Solving the equation of the area for the length l, noting that A=24, yields w=24/l.

    The domain for l and w is all positive reals.

    (1) There are an infinite number of possible perimeters. Start with any positive real number l; then, w is given by 24/l, and the perimeter will be 2l+2w.

    For example, if l=6, then w=4, and the perimeter is 20 meters. If l=13, then w=24/13, and the perimeter is 386/13.

    Another example would be the following: l=sqrt (2), w=24/sqrt (2), P=26sqrt (2)

    (2) Possible perimeters with integer (whole number) sides:

    1x24 P=50

    2x12 P=28

    3x8 P=22

    4x6 P=20

    These are the factorizations of 24.

    (3) There is no largest perimeter, but there is a least perimeter.

    We have P=2l+2 (24/l).

    Using calculus, we can find the minimum perimeter to occur when the length and width are both equal to 2sqrt (6).

    This gives a perimeter of 8sqrt (6) approx 19.5959.

    (4) We can graph the possible perimeters: P=2l+48/l
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