A rod of 108 meters long is bent to from a
rectangle . Find its dimensions if the area is maximum .

Explanation

Let x be the length and y be the breadth of the
rectangle .
Given ` 2x + 2y = 108 `
` rArr x + y = 54`
`rArr y = 54 - x `
Now , area of rectangle = xy
` = x (54 - x)`
` therefore f(x) = 54 x - x^(2)`
Differentiating f(x) w.r.t.x
` rArr f ' (x) = 54 - 2x `
Differentiating again w.r.t.x
` rArr f '' (x) = - 2 `
for extreme values ` f'(x) = 0 `
` therefore 54 - 2x = 0 `
` therefore x = 27 ` ltbr. ` f'' (27) = - 2 lt 0 `
` therefore ` Dimensions of rectangle are x = 27 and y = 54 - x = 27
` therefore ` Dimensions of rectangle are ` x = 27 m ` and
y = 27 m and it is a square.