# This is a math project for class 10.  Squares are cut from corners of a square sheet 10 cm and is then formed an open box. Find the size of squares cut when volume is maximum. So after...

This is a math project for class 10.

Squares are cut from corners of a square sheet 10 cm and is then formed an open box. Find the size of squares cut when volume is maximum.

So after calculations, the equation is reached how to find x.

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Let x be the length of the cut. Then the sides of the box formed are 10-2x by 10-2x by x. (x will be the "height" or depth of the open box; there are two corners for each side so the length of the side is 10-2x)

The volume of the box is l x w x h or:

V=(10-2x)(10-2x)x

You indicate that you are to proceed by guess and check: you should form a table of possible cuts and the associated volume:

Expanding the product we get `V=4x^3-40x^2+100x`

Note that there are some natural restraints on x (the domain of the problem): x must be greater than zero or no box is formed. Also x<5; if x=5 then there is no side left.

x      V
--------
0       0         Not possible but included for completeness
.25    22.5625
.5      40.5
.75    54.1875
1       64
1.25  70.3125
1.5    73.5
1.75  73.9375
2       72
2.25  68.0625
2.5    62.5
2.75  55.6875
3       48
3.25  39.8125
3.5    31.5
3.75  23.4375
4       16
4.25  9.5625
4.5    4.5
4.75  1.1875
5       0        Again impossible but included for completeness

Now the function `V=4x^3-40x^2+100x` has larger values, but they do make sense in the context of the problem.

It appears that the maximum occurs when x is about 2 3/4. You might try some more values in that area.

It can be shown with calculus that the maximum occurs when `x=5/3` with a maximum volume of `2000/27=74.bar(074)`

If you are allowed to use technology (a graphing utility or spreadsheet) you can pinpoint the maximum that way also.