# 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-...

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- Try same with different sized(10 cm, 30 cm, 40 cm, 50 cm, 100 cm ) square sheets of paper.

Q-Find a relationship (general rule) between the size of paper(y) and the size of cut(x) that produces the maximum volume?

Q- Test the validity of your general rule by using different values of a, b, and Justify your answer and its degree of accuracy.

Q- Discuss the scope or limitations of the general statement.

Q- Draw a graph Volume (V) and side of square (x) with the suitable scales.

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### 1 Answer

we have sheet 20 x 20 .let square of side x cut from each corner.Let it folded and form a box.

let V be the volume of the box. Then 2x length reduced from each side.

`V=(20-2x)^2xxx` , for getting max V ,we use calculus and calculate derivative

`(dV)/dx=2(20-2x)(-2)x+(20-2x)^2`

`=(20-2x)(-4x+20-2x)`

For max dv/dx=0 , (20-2x)(20-6x)=0

20-2x=0

x=10 ,not possible then there will no box

20-6x=0

x=10/3

max v=(20-20/3)^2(10/3)=16000/27 cubuc cm.