An open box is to be made out of a piece of a square card board of sides 18 cm by cutting off equal squares from the corners and turning up the sides. Find the maximum volume of the box.

Explanation

Let each side of the square cut off from each corner be x cm.
Then volume of box `V=(18-2x)(18-2x)x`
`V=(18-2x)^(2)x`
`V=4x^(3)+324x-72x^(2)" …(i)"`
Differentiating w.r.to x, we get
`(dV)/(dx)=12x^(2)+324-144x`
`(dV)/(dx)=12(x^(2)-12x+27)" ...(ii)"`
For maximum volume,
`(dV)/(dx)=0`
`rArr" "12(x^(2)-12x+27)=0`
`rArr" "x^(2)-9x-3x+27=0`
`rArr" "(x-9)(x-3)=0`
`rArr" "x=9,3`
Again differentiating, we get
`(d^(2)V)/(dx^(2))=2x-12" ...(iii)"`
at x = 9
`(d^(2)V)/(dx^(2))=+ve`
`therefore " V is minimum at x = 9"`
at x = 3
`(d^(2)V)/(dx^(2))=-ve`
`therefore" V is maximum at x = 3"`
`therefore" Maximum volume V "=(18-6)(18-6)xx3`
`" "=12xx12xx3=432cm^(3)`