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

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Discussion Overview

The discussion revolves around determining the dimensions of cut-out squares from a square sheet of metal to maximize the side area of a topless box. The problem involves understanding the setup of the equation and the interpretation of "side area," with a focus on mathematical reasoning and conceptual clarification.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant introduces the problem by defining the side length of the cut-out squares as "x" and notes the resulting dimensions of the box after folding the flaps.
  • Another participant calculates the total side area as $A = 4x(12-2x)$, suggesting a mathematical approach to finding the maximum area.
  • Some participants discuss the interpretation of "side area," with one suggesting it refers to the area of the sides, while another argues that different interpretations yield the same answer, indicating a level of ambiguity in the problem statement.

Areas of Agreement / Disagreement

Participants express differing interpretations of what "side area" means, leading to a lack of consensus on the terminology used in the problem. The discussion remains unresolved regarding the implications of these interpretations on the solution.

Contextual Notes

There is uncertainty about the definition of "side area" and its implications for the setup of the equation. The discussion does not resolve whether the area should be calculated for one side, all four sides, or in combination with the bottom of the box.

obesiston
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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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four sides, each with area $x(12-2x)$

therefore, total side area, $A = 4x(12-2x)$
 
I would be inclined to think that "side area" means "area of the sides" which is what what skeeter calculated.
 
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!
 

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