Answered

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

Sagot :

Numenius

Answer:

The perimeter is minimum for Length and width both are [tex]\sqrt3[/tex].

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)

[tex]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[/tex]

Now

[tex]\frac{d^2P}{dL^2}=\frac{12}{L^3}\\[/tex]

It is alays positive, so the perimeter is minimum.

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