Answer: 2 inches
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Work Shown:
x = length of the cut needed at each corner
Refer to the attached image below. Figure 1 shows a 18 by 12 rectangle. This is the original piece of cardboard. Figure 2 shows us cutting out red squares from each corner. Each red square is x inches by x inches, where x is the value we want to find. Note the resulting expressions 12-2x and 18-2x come from subtracting 2 copies of x for each dimension.
Figure 3 is the result of taking out the red squares. We will fold along the green dashed lines to get what you see in figure 4, which finally transitions to figure 5. This is the completed 3D box where there is no lid.
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The dimensions of this box are
length = 18-2x
width = 12-2x
height = x
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We want x to be some positive value, so x > 0
We also want the length to be positive, meaning that
length > 0
18-2x > 0
18 > 2x
2x < 18
x < 9
and we want the width to be positive as well, so,
width > 0
12 - 2x > 0
12 > 2x
2x < 12
x < 6
We find that x > 0 and x < 9 and x < 6.
Therefore, the range that x can be is 0 < x < 6. Keep this in mind when we get the three possible solutions below.
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Multiply out the length, width and height to get the volume 224
Length*Width*Height = volume of box
(18-2x)*(12-2x)*x = 224
x*(12-2x)*(18-2x) = 224
x*(4x^2-60x+216) = 224 ... use the FOIL rule
4x^3 - 60x^2 + 216x = 224 ... distribution
4x^3 - 60x^2 + 216x - 224 = 0 ... subtract 224 from both sides
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Now use a graphing calculator to graph y = 4x^3 - 60x^2 + 216x - 224
Use the root finder function on your calculator to find that the three roots are
x = 2
x = 2.73 (approximate)
x = 10.27 (approximate)
Which are the possible solutions; however, you'll find that x = 10.27 is not a valid solution as this x value is not in the range 0 < x < 6. In other words, x = 10.27 is too big a cut and not possible.
The only practical solutions are x = 2 or x = 2.73.
The easiest solution is x = 2 as there are no decimal or fractional values to worry about, so this is the best solution for Sabrina to go with.
She should make the cuts 2 inches.
