Question

A box with a square base and no top is to be built with a volume of 1638416384 in33. Find the dimensions of the box that requires the least amount of material. How much material is required at the minimum

Answer 1

`512\ \text{in}^2`

Explanation:

x = Length and width of base

y = Height of box

Volume of the box is
`16384\ \text{in}^3`

`x^2y=16384\\\Rightarrow y=(16384)/(x^2)`

Surface area is given by

`s=x^2+4y\\\Rightarrow s=x^2+4* (16384)/(x^2)\\\Rightarrow s=x^2+(65536)/(x^2)`

Differentiating with respect to x we get

`s'=2x-(131072)/(x^3)`

Equating with 0 we get

`0=2x^4-131072\\\Rightarrow x=((131072)/(2))^{(1)/(4)}\\\Rightarrow x=16`

`s''=2+(393216)/(x^4)`

at
`x=16`

`s''=2+(393216)/(16^4)=8>0`

So the function is minimum at x = 16

`y=(16384)/(x^2)=(16384)/(16^2)\\\Rightarrow y=64`

The material required is

`s=x^2+4y=16^2+4* 64\\\Rightarrow s=512\ \text{in}^2`

The minimum amount of material required is
`512\ \text{in}^2`.