Question

An open-top box with a square base is to have a volume of 4 cubic feet. Find the dimensions of the box (in ft) that can be made with the smallest amount of material.

Answer 1

The dimension of box=
`2ft* 2ft* 1ft`

Explanation:

We are given that

Volume of box=4 cubic feet

Let x be the side of square base and h be the height of box

Volume of box=
`lbh=x^2h`

`4=x^2h`

`h=(4)/(x^2)`

Now, surface area of box,A=
`x^2+4xh`

`A=x^2+4x((4)/(x^2))=x^2+(16)/(x)`

`(dA)/(dx)=2x-(16)/(x^2)`

`(dA)/(dx)=0`

`2x-(16)/(x^2)=0`

`2x=(16)/(x^2)`

`x^3=8`

`x=2`

`(d^2A)/(dx^2)=2+(32)/(x^3)`

Substitute x=2

`(d^2A)/(dx^2)=2+(32)/(2^3)=2+4=6>0`

Hence, the area of box is minimum at x=2

Therefore, side of square base,x=2 ft

Height of box,h=
`(4)/(2^2)=1 ft`

Hence, the dimension of box=
`2ft* 2ft* 1ft`