# A rectangul Length of rectangle sheet = 45 cm

Breath of rectangle sheet = 24 cm

Let the side length of each small square be a.

If by cutting a square from each corner and folding up the flaps we will get a cuboidal box with

Length, L = 45 – 2a

Breadth, B = 24 – 2a and

Height, H = a

Assuming,

Volume of box, V = LBH = (45 - 2a)(24 - 2a)(a)

Condition for maxima and minima is  (45 - 2a)(24 - 2a) + ( - 2)(24 - 2a)(a) + (45 - 2a)( - 2)(a) = 0 4a2 – 138a + 1080 + 4a2 – 48a + 4a2 – 90a = 0 12a2 – 276a + 1080= 0 a2 – 23a + 90= 0 a = 5, 18 = 24a - 276

For a = 5, = - 156,  For a = 18, = +156,  So for A to maximum Hence, a = 5 will give maximum volume.

And maximum volume V = (45 - 2a)(24 - 2a)(a) = 2450 cm3

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