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This is a math project for class 10. Squares are cut from corners of a square sheet...
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High School Teacher
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:
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.
0 0 Not possible but included for completeness
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.
Posted by embizze on May 22, 2013 at 3:11 PM (Answer #2)
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