Question

2. A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume of the box

Answer 1

`2 ft^3`

Explanation:

We are given that

Side of square shaped piece= 3ft

Let square of side x cut from each corner of square piece cardboard.

Length of box=
`3-2x`

Breadth of box=
`3-2x`

Height of box= x

We have to find the largest volume of box.

Volume of box is given by

`V=x(3-2x)^2`

`V=(4x^3-12x^2+9x)`

Differentiate w.r.t x

`(dV)/(dx)=12x^2-24x+9`

Substitute
`(dV)/(dx)=0`

`12x^2-24x+9=0`

`4x^2-8x+3=0`

`(2x-1)(2x-3)=0`

`2x-1=0\implies x=(1)/(2)`

`2x-3=0\implies x=(3)/(2)`

`x=(1)/(2), x=(3)/(2)`

Again differentiate w.r.t x

`(d^2V)/(dx^2)=24x-24`

Substitute
`x=(1)/(2)`

`(d^2V)/(dx^2)=12-24=-12<0`

Substitute
`x=(3)/(2)`

`(d^2V)/(dx^2)=36-24=12>0`

Therefore,
`(d^2V)/(dx^2) <0` at
`x=(1)/(2)`

Hence, the volume is maximum at
`x=(1)/(2)`

Substitute the value
`x=(1)/(2)` then we get

`V=(1)/(2)(3-1)^2=2 ft^3`