Expert: Scott A Wilson - 10/14/2009

Question
An open box is to be made from a flat piece of material 11 inches long and 6 inches wide by cutting equal squares of length xfrom the corners and folding up the sides.
Write the volume V of the box as a function of x. Leave it as a product of factors, do not multiply out the factors.

If we write the domain of the box as an open interval in the form (a,b), then what is a and b?



I already got the answer for the first part: 4x^3 - 34x^2 + 66
The only thing I have trouble is finding the domain. For 'a' I got 11/2, but I don't know how to figure out 'b.'Thanks for the help.  

Answer
The domain is what values of x are possible.  The problem is right: nothing needs to be done.
Not that the paper is 11x6.  Since x is cut out of each corner, this means 2x is cut out of each side.  If we take one side, note that the amount left is 11-2x, so x needs to be between 0 and 5.5.  If we look at the other side, it is even more restrictive, for to have a side of 6 - 2x, we need that to be positive, so 6 >= 2x, so x is at most 3.
The domain is then (0,3).  Note that if x=0 or x=3, it makes no sense,
since it is not really a box, but only a flat sheet.



Now about the bottom, that answer you got is almosst right.
The answer to that is really 4x^3 - 34x^2 + 66x.
This can be factored to 2x(2x² - 17x + 33) at first.

The quadratic equation can then be applied, so for 2x² - 17x + 33, we have a=2, b=-17, and c=33.
The quadratic equation is (-b±√(b²-4ac))/(2a).

That gives (17±√(289-264))/4 = (17±√25)/4 = (17±5)/4,
so the two solutions here are 22/4 and 12/4.

That is the same as 11/2 and 3.

The equation is then x(x - 11/2)(x-3).
That was fun, and was done just since I noticed the formula wasn't quite correct.

Remember, the height was x and we get height(width)length, so each term has an x in it.