Question

A box with a square base and no top is to be built with a volume of 4000 in3. find the dimensions of the box that requires the least amount of material. how much material is required at the minimum?

Answer 1

Let the length of the square base be x inches .

And the height of the box be y inch .

Volume = x*x*y

4000 = x *x *y

`y = (4000)/(x^2)`

S = x*x + 2xy + 2 x y

`S = x^2 + 4 xy`

`S = x^2+ 4x* (4000)/(x^2)`

`S = x^2 + (16000)/(x)`

`S ' = 2x - (16000)/(x^2)`

S ' =0

`2x - (16000)/(x^2)=0`

`2x^3 - 16000 = 0`

`image`

x = 20

`y = (4000)/(20^2) = (4000)/(400) = 10`