Question

How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of `8,788 (cm)^3`?

Answer 1

You set `x` as being the sides, and `h` for the height.

The box will have a square bottom.
Then the amount of cardboard used will be:
For the bottom: `x*x=x^2`
For the sides: `x*h*4`(sides)`=4xh`

Total area : `A=x^2+4xh`

The volume of the box= `x*x*h=8788` from which we can conclude that `h=8788/x^2`

Substituting that into the formula for the area `A`, we get:

`A=x^2+4x*(8788/x^2)=x^2+35152/x`

To find the minimum, we have to differentiate and set to `0`

`A'=2x-35152/x^2=0->2x=35152/x^2` multiply by `x^2`

`2x^3=35152->x^3=17576->x=root 3 17576=26`
Substitute: `h=8788//26^2=13`

Answer :
The sides will be `26cm` and the height will be `13cm`