Question

An open-top box with a square base is to be constructed from 120 square centimeters of material. What dimensions will produce a box with the largest possible volume

Answer 1

Maximum length and breadth of the box is 6.32 cm and the height of the box is 3.16 cm

Explanation:

Surface area of box

`4xy+x^2=120\\\Rightarrow y=(120-x^2)/(4x)`

Volume of box is

`V=x^2y\\\Rightarrow V=x^2* (120-x^2)/(4x)\\\Rightarrow V=(120x-x^3)/(4)\\\Rightarrow V=30x-(1)/(4)x^3`

Differentiating with respect to
`x`

`(dV)/(dx)=30-(3)/(4)x^2`

Equating with zero

`30-(3)/(4)x^2=0\\\Rightarrow x^2=(30* 4)/(3)\\\Rightarrow x^2=40\\\Rightarrow x=6.32`

Double derivative of the volume

`(d^2V)/(dx^2)=-(6x)/(4)=-(6* 6.32)/(4)=-9.48<0`

So, the volume is maximum at
`x=6.32`

`y=(120-x^2)/(4x)=(120-40)/(4* √(40))\\\Rightarrow y=3.16`

So, the maximum length and breadth of the box is 6.32 cm and the height of the box is 3.16 cm.