A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top by cutting off squares from the corners and folding up the flaps. What should be the side of the square in order the volume of the box is maximum.

Explanation

Let the square of side 4 cm is cut form four corners. Now constructing a box by folding the corner, then length, breadth and height of this box are (45 - 2x) cm, (24- 2x) cm and x cm respectively.
Volume of box , `V = (45- 2x)(24 - 2x)*x`
` = 4x^(3) - 138 x^(2) + 1080 x`
` rArr (dV)/(dx) = 12x^(2) - 276x + 1080`
For maxima/minima ` (dV)/(dx) = 0`
` rArr 12(x^(2) - 23x + 90) =0`
` rArr ( x- 5)(x-18) = 0`
` rArr x = 5 or x = 18`
x = 18 cm is not possible because at x = 18
24 - 2x = 24 - 36 =- 12
which is impossible.
` :. x = 5 cm `
` ((d^(2)V)/(dx^(2))) = 24x - 276`
at ` x = 5, ((d^(2)V)/(dx^(2))) = 120 - 276 =- 156 lt 0 `
`rArr ` at x = 5 , volume is maximum.
`:.` Required side of square = 5 cm.