Dimension of the cut-out squares that result in largest possible side area

In summary, the conversation discusses creating a topless square box by cutting squares out of the corners of a 12-inch square sheet of metal and folding up the resulting flaps. The goal is to find the largest possible side area, which can be represented as 4x(12-2x) where x is the length of the cut-out squares. Both the area of one side and the total side area would yield the same answer.
  • #1
obesiston
1
0
A topless square box is made by cutting little squares out of the four corners of a square sheet of metal 12 inches on a side, and then folding up the resulting flaps. What is the largest side area which can be made in this way?

What information I have so far is that since the side of the little squares are unknown, I called them "x", and so since the length of the full box is 12 inches, once folded up I'd have 12-2x. One thing I'm having trouble with is setting up my equation. I've done a similar problem before where it asks for volume, but I don't fully understand what it means by "side area." I also need to complete the perfect square to find dimension of the cut-out squares that result in largest possible side area.
 
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  • #2
four sides, each with area $x(12-2x)$

therefore, total side area, $A = 4x(12-2x)$
 
  • #3
I would be inclined to think that "side area" means "area of the sides" which is what what skeeter calculated.
 
  • #4
If you interpret "largest side area" as meaning the area of one side, instead of all four, or the area of the four sides plus the bottom, you get the same answer so it really doesn't matter!
 

FAQ: Dimension of the cut-out squares that result in largest possible side area

What is the "largest possible side area"?

The largest possible side area refers to the maximum area that can be obtained by cutting out squares from a given shape. It is the result of finding the optimal dimensions for the cut-out squares.

How do you determine the dimensions of the cut-out squares for the largest possible side area?

The dimensions of the cut-out squares can be determined by finding the square root of the total area of the given shape and dividing it by the number of cut-out squares. This will result in equal dimensions for each cut-out square, which will maximize the side area.

Can the dimensions of the cut-out squares vary depending on the shape?

Yes, the dimensions of the cut-out squares can vary depending on the shape. The optimal dimensions for the cut-out squares will differ for different shapes, as the total area and number of cut-out squares will vary.

Is the largest possible side area always the most efficient option?

Not necessarily. While the largest possible side area may result in the maximum area, it may not always be the most efficient option. Other factors such as material cost and structural stability should also be considered when determining the dimensions of the cut-out squares.

Are there any other methods for determining the dimensions of the cut-out squares?

Yes, there are other methods such as using mathematical equations or computer algorithms to find the optimal dimensions for the cut-out squares. These methods may be more complex but can result in more precise and efficient solutions.

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