SOLUTION: I need help solving this word problem: A box with no lid is to be constructed out of a piece of cardboard 17'' by 22'' by cutting out squares x'' per side in each corner and then

Question 728575: I need help solving this word problem:
A box with no lid is to be constructed out of a piece of cardboard 17'' by 22'' by cutting out squares x'' per side in each corner and then folding up the sides. If the volume of the resulting box is to be 152.5 cubic inches, find the sizes of the cut-outs. Round your answear(s) to nearest hundredth. What is your domain?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Cut along red lines; fold along blue lines.
square inches.
The box has a height of inches.
The volume of the box is
cubic inches.

The expression above gives you the volume as a function of , but the fact that is the measure of the side of the squares cut off imposes some restrictions:
and --> --> --> or
So the variable must comply with

That is the domain of the function .

cubic inches, so we must solve
and find one or more solutions with .
That is a cubic (degree 3) equation, and may be difficult to solve.
There could be a rational solution somewhere that could be used to reduce the equation to a quadratic equation.
Otherwise, if the solutions turn out to be irrational, it is a question of finding approximate solutions by painful calculations. The request to round answers to the nearest hundredth suggests that is the case.
A graphing calculator, or some computer software could help with those calculations. The question about the domain suggest that using a graphing calculator is expected. If you have been using a graphing calculator in class, that must be what is expected.
The polynomial function would be zero at x=0 and x=8.5, and we know that it is positive in between.
Logic tells us that has a maximum in the domain .
For --> , so 152.5 is not the maximum.
We should expect two solutions, one on each side of the maximum.
I find that there is a solution between 0.445 and 0.450, which would be rounded to the nearest hundredth.
There is another solution between 7.115 ans 7.12 which would round to
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