# Topic: Math 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...

## Topic: Math

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?

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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` ``

`=(y-2x)(-4x+y-2x)`

For max volume ,dv/dx =0

(y-2x)(y-6x)=0

y=2x not possible ,otherwise there will no box.

y-6x=0

y=6x

x=y/6

`(d^2v)/(dx^2)=-2(y-6x)+(y-2x)(-6)` ``

`{(d^2v)/(dx^2)}_{x=y/6}=(y-(2y)/6))(-6)`

=-4y

`{(d^2v)/(dx^2)}_{x=y/6}=-4y <0`

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`