Question

A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.

Answer 1

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