Ineed this asap
sabrina is making an open box from a piece of cardboard that has a width
ma piece of cardboard that has a width of 12 inches and a length
of 18 inches. she'll form the box by making cuts at the corners and folding up the
y making cuts at the corners and folding up the sides. if she wants the
box to have a volume of 224 in3, how long should she make the cuts?

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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.