Question

Find the area of the largest rectangle having the perimeter of 200 metres.

Solution

Let x, y be the lengths of sides of rectangle having perimeter of 200 metres.
$therefore space space 2 straight x plus 2 straight y space equals 100 space space rightwards double arrow space space space space straight x plus straight y equals space 50$
$therefore$ $straight y space equals space 50 minus straight x$ ...(1)
Let $straight A space equals space xy space equals straight x left parenthesis 50 minus straight x right parenthesis space equals space 50 straight x minus straight x squared$ $open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$
$dA over dx space equals space 50 minus 2 straight x$
$dA over dx space equals space 0 space space space space give space us space 50 minus 2 straight x space equals space 0 space space space space space space rightwards double arrow space space space straight x space equals space 25$
$fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space minus 2$
$At space space straight x space equals 25 comma space space space fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space minus 2 space less than space 0$
$therefore space space space space straight A space is space maximum space when space straight x space equals space 25 comma space space space straight y space equals space 25 therefore space space space area space of space largest space rectangle space equals 25 space cross times space 25 space equals space 625 space sq. metres.$

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Application Of Derivatives

Find the area of the largest rectangle having the perimeter of 200 metres.

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