How Can You Optimize Material in Manufacturing Open-Top Square Base Boxes?

[Lurid]

Homework Statement

Suppose that you are the manager of a sheet metal shop. A customer asks you to manufacture 10,000 boxes, each box being open on the top. The boxes are required to have a square base and a 9 cubic foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of metal and folding along the edges. What are the dimensions of the square to be cut if the area of the square piece of sheet metal is 100 square feet?

Homework Equations

_|____|_
..|...|
_|____|_
x|...|

Let x = the dimension of the square you cut out. Then the dimension of the inner square is (10 - 2x).
The box has a base of (10 - 2x) by (10 - 2x) and a height of x.

The Attempt at a Solution

(10-2x)²x=9
4x^3-40x^2+100x-9=0

I'm not sure if I'm even interpreting this problem correctly, because I can't even factor it. I think the part that's throwing me off is the "10,000 boxes" that must be made. I'm not sure how to add it into the equation. Any help is appreciated!

 

[LCKurtz]

Homework Statement

Suppose that you are the manager of a sheet metal shop. A customer asks you to manufacture 10,000 boxes, each box being open on the top. The boxes are required to have a square base and a 9 cubic foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of metal and folding along the edges. What are the dimensions of the square to be cut if the area of the square piece of sheet metal is 100 square feet?

Homework Equations

_|____|_
..|...|
_|____|_
x|...|

Let x = the dimension of the square you cut out. Then the dimension of the inner square is (10 - 2x).
The box has a base of (10 - 2x) by (10 - 2x) and a height of x.

The Attempt at a Solution

(10-2x)²x=9
4x^3-40x^2+100x-9=0

I'm not sure if I'm even interpreting this problem correctly, because I can't even factor it. I think the part that's throwing me off is the "10,000 boxes" that must be made. I'm not sure how to add it into the equation. Any help is appreciated!

You have left out a critical piece of information. Surely the problem must require that your boxes use the minimum amount of metal. You need to write the equation for what you are trying to minimize.

 

[Ray Vickson]

Homework Statement

Suppose that you are the manager of a sheet metal shop. A customer asks you to manufacture 10,000 boxes, each box being open on the top. The boxes are required to have a square base and a 9 cubic foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of metal and folding along the edges. What are the dimensions of the square to be cut if the area of the square piece of sheet metal is 100 square feet?

Homework Equations

_|____|_
..|...|
_|____|_
x|...|

Let x = the dimension of the square you cut out. Then the dimension of the inner square is (10 - 2x).
The box has a base of (10 - 2x) by (10 - 2x) and a height of x.

The Attempt at a Solution

(10-2x)²x=9
4x^3-40x^2+100x-9=0

I'm not sure if I'm even interpreting this problem correctly, because I can't even factor it. I think the part that's throwing me off is the "10,000 boxes" that must be made. I'm not sure how to add it into the equation. Any help is appreciated!

The 10,000 boxes have nothing to do with the problem---as it was stated. You need to solve a cubic equation; there are formulas for doing that, but using a numerical method is easier. There are three solutions; one of them has 2*x > 10 so is not usable; the other two lead to feasible, but ridiculous boxes (either very tall and skinny or very short and fat).

BTW: your title "systems of linear equations" is highly misleading: you don't have a system of equations, and your equation is not linear.

RGV

 

[Lurid]

The 10,000 boxes have nothing to do with the problem---as it was stated. You need to solve a cubic equation; there are formulas for doing that, but using a numerical method is easier. There are three solutions; one of them has 2*x > 10 so is not usable; the other two lead to feasible, but ridiculous boxes (either very tall and skinny or very short and fat).

BTW: your title "systems of linear equations" is highly misleading: you don't have a system of equations, and your equation is not linear.

RGV

I'm still kinda confused. I have a cubic equation, but I can't simplify it any further. It also doesn't really make sense to me theoretically speaking, because how can you make a box that is 9 cubic feet with a 100 square feet sheet metal? Is it even possible? I'm still kinda convinced that my method is wrong.

Ah, sorry if it's misleading, but it's a problem under the section "Systems of Nonlinear Equations" in my math textbook. I guess it would be fitting?

Edit:
Oops! It's supposed to be 'nonlinear', not 'linear.
And here's the problem from my textbook, if it makes any difference.
http://i49.tinypic.com/21m6whz.jpg

 

[HallsofIvy]

Read Ray Vickson's response again, particularly:
"You need to solve a cubic equation; there are formulas for doing that, but using a numerical method is easier. There are three solutions; one of them has 2*x > 10 so is not usable; the other two lead to feasible, but ridiculous boxes (either very tall and skinny or very short and fat). "

 

[Ray Vickson]

I'm still kinda confused. I have a cubic equation, but I can't simplify it any further. It also doesn't really make sense to me theoretically speaking, because how can you make a box that is 9 cubic feet with a 100 square feet sheet metal? Is it even possible? I'm still kinda convinced that my method is wrong.

Of course it is possible; solving the cubic equation will tell you how to do it.



Ah, sorry if it's misleading, but it's a problem under the section "Systems of Nonlinear Equations" in my math textbook. I guess it would be fitting?

Edit:
Oops! It's supposed to be 'nonlinear', not 'linear.
And here's the problem from my textbook, if it makes any difference.
http://i49.tinypic.com/21m6whz.jpg

Part (b) leads to a much more interesting problem, but one that requires calculus methods to find the best design. (Just for the record, the solution would be to use square sheets of area 24.53112800 sq. ft.--which is a square with sides of length 4.952890873 ft--and to cut out corners of length 0.8254818122 ft. This would minimize the total area of all sheets used.)

RGV