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How do you find the length and width of a rectangle whose area is 900 square meters and whose perimeter is a minimum?

1 Answer

Jun 11, 2015

Length = Width = 30 meters

Explanation:

If $L \times W = 900$ $LxxW = 900$
$\Rightarrow L = \frac{900}{W}$ $rArr L= 900/W$

$P = 2 \cdot (L + W)$ $P = 2*(L+W)$

$XX$ $color(white)("XX")$ $= 2 (\frac{900}{W} + W)$ $= 2(900/W +W)$

$XX$ $color(white)("XX")$ $= 1800 W^{- 1} + 2 W$ $=1800W^(-1) + 2W$

The minimum will occur when $\frac{d P}{d W} = 0$ $(dP)/(dW) =0$

$\frac{d P}{d W} = - 1800 W^{- 2} + 2 = 0$ $(dP)/(dW) = -1800W^(-2) +2 = 0$

$XXXX$ $color(white)("XXXX")$ 1/W^2 = (-2)/(-1800)#

$XXXX$ $color(white)("XXXX")$ $W^{2} = 900$ $W^2 = 900$

$XXXX$ $color(white)("XXXX")$ $W = 30$ $W =30$

and since $L \times W = 900$ $LxxW =900$
$\Rightarrow L = 30$ $rArr L =30$

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