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Question
A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.
Solution
Let the length and breadth of a rectangle be l and b.
∴ Perimeter of rectangle = 2 (l + b) = 36cm
∴ l + b = 18 ....(i)
Area of rectangle = l × b = l (18 - l)
Let f(l) = 18l - l2
∴ f'(l) = 18 - 2l
and f''(l) = - 2
Consider, f '(l) = 0
∴ 18 - 2l = 0
∴ 18 = 2l
∴ l = 9
For l = 9,
f''(9) = - 2 < 0
∴ f(x), i.e. area is maximum when l = 9 cm
and b = 18 - 9 ....[From (i)]
= 9 cm
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