SOLUTION: Can you please explain to me how to do this?? Mr. Greene has 8.5-by-11 in. cardboard sheets. As a class project, Mr. Greene asked each of his students to make an open-top box un

Question 255965: Can you please explain to me how to do this??
Mr. Greene has 8.5-by-11 in. cardboard sheets. As a class project, Mr. Greene asked each of his students to make an open-top box under these conditions:
I) Each box must be made by cutting small squares from each corner of a cardboard sheet.
II) The box must have a volume of 48 in^3.
III) The amount of cardboard waste must be minimized.
What is the approximate side length for the small squares that would be cut from the cardboard sheet?
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Mr. Greene has 8.5-by-11 in. cardboard sheets. As a class project,
Mr. Greene asked each of his students to make an open-top box under these conditions:
:
I) Each box must be made by cutting small squares from each corner of a cardboard sheet.
II) The box must have a volume of 48 in^3.
III) The amount of cardboard waste must be minimized.
What is the approximate side length for the small squares that would be cut from the cardboard sheet?
:
Draw a rectangle representing the 8.5 by 11 in sheet. Cut out a small square in each corner, label the side of the square as x.
:
Visualize bending the sides up to form an open box; it will be apparent that
Box dimensions will be:
(11-2x) = box length
(8.5-2x) = box width
x = box height
:
Vol: height * Length * width
x*(11-2x)*(8.5-2x) = 48
FOIL
x(93.5 - 22x - 17x + 4x^2) = 48
x(93.5 - 39x + 4x^2) = 48
93.5x - 39x^2 + 4x^3 - 48 = 0
The equation
4x^3 - 39x^2 + 93.5x - 48 = 0
:
Now the easiest way, is to plot this equation on a graphing calc

Note we have 3 solutions for x, the last one x=6+ is not valid but
x = .7 and = 2.7 are, however, the removed square would be "waste" therefore
x = .7 would be the smallest square, that would be the choice

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