Question

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

Answer 1

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.