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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. Find the maximum volume of the box.

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Let x be the side of the square which is to be cut off from each corner.

Then dimensions of the box are 18 - 2x, 18 - 2x and x.

Let V be the volume of the box then

Also,

For maximum or minimum value V' = 0

⇒ (18 -  2x) (- 6) = 0

⇒ x = 9

or x = 3

Neglecting x = 9

∵ For x = 9,length = 18 - 2x = 18 - 2(9) = 0]

Therefore, x = 3 is to be taken.

V'(3) = (18 - 6)(- 6) + (18 - 18) (- 2) = 72 < 0

Thus, volume is maximum when x = 3

∴ Length = 18 - 2x = 18 - 6 = 12 cm;

Breadth = 18 - 2x = 18 - 6 = 12cm;

Height = x = 3 cm.

Maximum volume of the box,

= 12 x 12 x 3 = 432cm3

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