Question

An open box (i.e. without a lid) is built in the shape of a cuboid to minimise the total

Answer 1

The lenght of the base is four times the width of the base, so the volume must be:

`V=l\cdot w\cdot h`
`400cm^3=4x\cdot x\cdot h`
`h=(400)/(4x^2)`
`h=(100)/(x^2)`

For the surface area we have, the area of the base:

`A_b=4x\cdot x`
`A_b=4x^2^{}`

the area of 2 sides must be:

`2A_s=2\cdot w\cdot h`
`2A_s=2\cdot x\cdot(100)/(x^2)`
`2A_s=(200)/(x)`

and for the other 2:

`2A_(s2)=2\cdot l\cdot h`
`2A_(s2)=2\cdot4x\cdot(100)/(x^2)`
`2A_(s2)=(800)/(x)`

And adding all we have:

`A=A_b+2A_s+2A_(s2)`
`A=4x^2+(200)/(x)+(800)/(x)`
`A=4x^2+(1000)/(x)`