TASK: Investigation with square sheets: From a square sheet of paper 20 cm by 20 cm, we can make a box without a lid. We do this by cutting a square from each corner and folding up the flaps.
Q- What is the maximum possible volume and what size cut produces it?
Q-Find a relationship (general rule) between the size of paper(y) and the size of cut(x) that produces the maximum volume?
Let side of square paper be `yxxy`
let from each corner cut square paper of size `x xx x`
So legth defected paper will be (y-2x)
Let box prepared with this length and it volume be V
`V= (y-2x)^2 xx x`
`(dv)/(dx)= 2(-2)(y-2x)x+(y-2x)^2` ``
For max volume ,dv/dx =0
y=2x not possible ,otherwise there will no box.
so x=y/6 give max volume.
General relation between side and volume will be
`V=(y-2x)^2 xx x ,y!=2x`
Substitute y=20 ,you can get a particular relation
`V=(20-2x)^2 xx x`