SOLUTION: You plan to make an open box by cutting equally sized squares from each corner of a 10-inch by 12-inch rectangular piece of material and then folding up the flaps. This is il

Question 1055329: You plan to make an open box by cutting equally sized squares
from each corner of a 10-inch by 12-inch rectangular piece of
material and then folding up the flaps. This is illustrated below.
You need the open box to contain 80 cubic inches of liquid. You
can do this by removing corners that are each 1 inch by 1 inch.
However, the box will then be too wide to fit where you need to
put it. Find a differently sized square to remove from each
corner so that the volume of the box is still 80 cubic inches.
Answer by Edwin McCravy(20062) (Show Source): You can put this solution on YOUR website!

Cut out a square of x inches by x inches out of each 
side of the length and width, which leaves flaps of 
10-2x inches and 12-2x inches and when they are folded 
up the height of the box will be x:
    ______
 __|______|__
|  |      |  |10-2x
|__|______|_x|
   |______|
     12-2x

Use the Volume formula

LWH = V

(12-2x)(10-2x)x = 80

Multiply that out and get

4x^3-44x^2+120x-80 = 0

Divide through by 4

x^3-11x^2+30x-20 = 0

We are told that x=1 is a solution.  So we use synthetic
division to factor it:

1 | 1 -11  30 -20
  |     1 -10  20
    1 -10  20   0

So the left side of

x^3-11x^2+30x-20 = 0

factors as

(x-1)(x^2-10x+20) = 0

The quadratic doesn't factor so we solve it by
the quadratic formula and get the solutions



Using the - we have  which is about 2.7639

Using the - we have  which is about 7.2361

We have to ignore the 2nd answer because if we cut out a square
that big we could not make a box, because 7.2361 is more than
half of the 10 inch side.

Answer: cut out a square of size  by .
or 2.7639 inches by 2.7636 inches.

Edwin

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