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
Side length of big square is 18 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 = 18 – 2a
Breadth, B = 18 – 2a and
Height, H = a
Assuming,
Volume of box, V = LBH = a(18 - 2a)2
Condition for maxima and minima is
(18 - 2a)2 + (a)(- 2)(2)(18 - 2a) = 0
(18 - 2a)[(18 - 2a) - 4a] = 0
(18 - 2a)[18 – 6a] = 0
a = 3, 9
( - 2)(18 – 6a) + ( - 6)(18 – 2a)
For a = 3, 
= - 72, 
For a = 9, 
= 72, 
So for A to maximum
Hence, a = 3 will give maximum volume.
And maximum volume, V = a(18 - 2a)2 = 432 cm3