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.