A Rectangle Has Area 50 Cm2 . Find Its Dimensions When Its Perimeter is the Least - Mathematics and Statistics

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A rectangle has area 50 cm2 . Find its dimensions when its perimeter is the least

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Let x cm and y cm be the length and breadth of the rectangle.Then its area is xy = 50


Perimeter of the rectangle= 2(x + y)

= `2(x+50/x)`

`Let f(x)=2(x+50/x)`

`then  f'(x)=2(1-50/x^2) and f''(x)=2(100/x^3)`

`if f'(x)=0,then 2(1-50/x^2)=0`



But x can not be negative and hence `x =5sqrt2 `

`and f''(5sqrt2)=200/(5sqrt2)^3>0`

f has a minimum value at `x = 5sqrt2`

`For x=5sqrt2, y=50/(5sqrt2)=5sqrt2`


Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
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2014-2015 (October)



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