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]
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
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.
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)
To find the size of the corner squares that should be cut, we need to first determine the dimensions of the box that will result in a volume less than 235 cubic feet.
Let's assume that the side length of each corner square to be cut is 'x'. When the flaps are folded up, the dimensions of the resulting box will be:
Length: 22 - 2x
Width: 27 - 2x
Height: x
The volume of the box can be calculated by multiplying these dimensions:
Volume = (22 - 2x)(27 - 2x)(x)
To find the size of the corner squares that yields a volume less than 235 cubic feet, we need to solve the inequality:
(22 - 2x)(27 - 2x)(x) < 235
Now, we can solve this inequality by finding the values of 'x' that satisfy it. However, since the equation is a polynomial and involves multiple factors, it's best to use a graphing calculator or software to visualize the solution.
From your response, it seems that you have already calculated the values for 'x'. If those values are indeed correct, you can use either brackets or parentheses to represent the solution interval. However, it is important to clearly define the range of values for 'x' that satisfy the inequality.
Based on the values you provided (0 and 0.42), it seems that the range of 'x' for a box with a volume less than 235 cubic feet is [0, 0.42]. However, without further information, it is difficult to confirm whether 9.46 is the upper limit for 'x' or if there was a calculation error.
To be absolutely sure, it is best to double-check your calculations using a graphing calculator or software. This way, you can accurately determine the size corner squares that should be cut to yield a box with a volume less than 235 cubic feet and correctly represent the solution interval using either brackets or parentheses.