Find the dimensions of a rectangle with area 3 square meters whose perimeter is as small as possible.

Question

Find the dimensions of a rectangle with area 3 square meters whose perimeter is as small as possible.

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Cherry 3 years 2021-08-05T18:27:04+00:00 1 Answers 9 views 0

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  1. dangkhoa
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    2021-08-05T18:28:57+00:00
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    Answer:

    The perimeter is minimum for Length and width both are \sqrt3.

    Step-by-step explanation:

    Area, A = 3 square metre

    Let the length is L and width is W.

    Area = L W

    3 = L W…..(1)

    The perimeter is given by

    P = 2 (L + W)

    Substitute the value of  from (1)

    P = 2 \left ( L +\frac{3}{L} \right )\\\\P = 2 L + \frac{6}{L}\\\\\frac{dP}{dL} = 2 - \frac{6}{L^2}\\\\Now\\\\\frac{dP}{dL}=0\\\\2 - \frac{6}{L^2} = 0\\\\L = \sqrt 3, W = \sqrt 3

    Now

    \frac{d^2P}{dL^2}=\frac{12}{L^3}\\

    It is alays positive, so the perimeter is minimum.

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