SOLUTION: 2. The members of a group of packaging designers of a gift shop are looking for a precise procedure to make an open rectangular box with a volume of 560

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Question 988552: 2. The members of a group of packaging designers of a gift shop are looking for a precise procedure to make an open rectangular box with a volume of 560 cubic inches from a 24-inch by 18-inch rectangular piece of material. The main problem is how to identify the side of identical squares to be cut from the four corners of the rectangular sheet so that such box can be made
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
w and L are the KNOWN width and length of the rectangular material. x is the side length of each square to remove to form the open top box. x is UNKNOWN. The volume V of the box is KNOWN.

You would have system%28V=560%2CL=24%2Cw=18%29.

The boxes base area is %28w-2x%29%28L-2x%29.
The height of the box is x.
The volume will be highlight_green%28x%28w-2x%29%28L-2x%29=V%29.

Begin simplifying.
x%28wL-2Lx-2wx%2B4x%5E2%29-V=0
4x%5E3-2Lx%5E2-2wx%5E2%2BwLx-V=0
This being a cubic equation very likely to be handled with Rational Roots Theorem or maybe graphing software, substituting the known values and simplifying the equation with those calculations is the next thing to do.

You can continue this from here hopefully.

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

2. The members of a group of packaging designers of a gift shop are looking for a precise procedure to make an open rectangular box with a volume of 560 cubic inches from a 24-inch by 18-inch rectangular piece of material. The main problem is how to identify the side of identical squares to be cut from the four corners of the rectangular sheet so that such box can be made
Let side of one square to be cut be S

Then height of box = S inches

Assuming length of material is 24 inches, the length of the box, after squares have been cut = 24 - 2S

Assuming width of material is 18 inches, the width of the box, after squares have been cut = 18 - 2S

Volume needed: 560 cub inches

We now have volume of box as: S(24 - 2S)(18 - 2S) = 560

S%28432+-+84S+%2B+4S%5E2%29+=+560+

432S+-+84S%5E2+%2B+4S%5E3+=+560 

4S%5E3+-+84S%5E2+%2B+432S+-+560+=+0

4%28S%5E3+-+21S%5E2+%2B+108S+-+140%29+=+4%280%29

S%5E3+-+21S%5E2+%2B+108S+-+140+=+0

Using the rational root theorem, we get 2 as one of the zeroes. Therefore, S or side of one of the 4 squares to be cut off = highlight_green%282%29 inches