Question 984998
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If you cut *[tex \Large x] by *[tex \Large x] out of each corner and fold up the sides, the measure of the sides of the square bottom of the box will be *[tex \Large 20\ -\ 2x] by *[tex \Large 20\ -\ 2x] and the measure of the depth of the box will simply be *[tex \Large x].  So:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ 4x^3\ -\ 80x^2\ +\ 400x]


but this function has a restricted range because any value of *[tex \Large x] smaller than zero or larger than 10 is absurd, while *[tex \LARGE x\ =\ 0] or *[tex \Large x\ =\ 10] results in a zero volume box.  Given these restrictions, and the problem requirement of *[tex \Large x\ \in\ \mathbb{Z}], the possible answers are one of the numbers 1 through 9 inclusive.  You could try each of these nine possibilities in the volume function to see which one gives you the largest answer, or you could look at a graph of the function and perhaps narrow your search somewhat.


*[illustration TurnedUpBox].


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \