Question

An open box is formed by cutting squares out of a piece of cardboard that is 22 ft by 27 ft and folding up the flaps.

What size corner squares should be cut to yield a box that has a volume of less than 235 cubic feet?

I know that the size corner squares are .42 and 9.46. My teacher wants either brackets or parenthesis. I know if it's more than it gets parenthesis and at least gets brackets. I'm not sure what to do if it is less than.

Would it be:

[0,.42]union[.42,9.46]

Answer 1

Your equation would be
x(20-2x)(27-2x) < 235 or
4x^3 - 98x^2 + 594x - 235 < 0

( I am curious what method you used to solve x = .425 and x = 9.46)

If Volume = 4x^3 - 98x^2 + 594x - 235
we get a cubic, which has Volume = 0 at
x = .425, 9.46, and 14.6

we get a positive Volume for x's between .425 and 9.46, all other values of x produce negative volumes, (which makes no sense)

All this square bracket and round bracket stuff is new to me, in my days we would have simply said:
.425 < x < .946

Answer 2

I used the same equation you said then put it in the graphing calculator and found the intersection points. I'm thinking that you would put
[.42,9.46) to show that it is more than .42 and less than 9.46.

Answer 3

I believe in the "interval notation"
a [ means you include the number,
while ( excludes the number

so [.42,9.46)
would mean .42 ≤ x < 9.46

are you sure you want to include the .42 ?
It produces a volume of zero.
I know that is less than 235
but so would all negative volumes obtained by x's outside my domain.

I think I would go with (.42 , 9.46)