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A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Also, find this maximum volume

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Given as the side length of big square is 18 cm

Suppose the side length of each small square be a.

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

Length, L = 18 – 2a

Breadth, B = 18 – 2a and

Height, H = a

Assuming, the volume of box, V = LBH = a (18 – 2a)2

Condition for the maxima and minima is

Thus, a = 3 will give maximum volume.

And the maximum volume, V = a(18 - 2a)2 = 432 cm3

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