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A square piece of tin of side 12 cm is to be made into a box without a lid by cutting a square from each corner and folding up the flaps to form the sides. 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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Consider x cm as the length cut off from each corner

So the length of box is (12 – 2x) cm and x cm as its breadth and height

Here volume can be written as

V = (12 – 2x)2 × x

On further simplification

V = 4x3 – 48x2 + 144x

By differentiating w.r.t. x

We know that V is maximum when x = 2

Here V = {(12 – 4)2 × 2} = {64 × 2} = 128 cm3

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