Optimization problem: Folding a triangle to minimize one side

In summary, to minimize the length of the fold when folding the upper right-hand corner of a 12 in by 8 in piece of paper to the bottom edge, one would need to relate the length of the fold (y) to the length of the unfolded side (x) and then differentiate to find the minima. Using the information of the paper's dimensions, we can determine that the bottom piece after folding would be 8-x in length. Additionally, the angles in the two triangles formed by the fold would be related in a way that can help in determining the optimal length of the fold.
  • #1
PsychStudent
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Homework Statement



The upper right-hand corner of a piece of paper, 12 in by 8 in is folded over to the bottom edge. How would you fold it to minimize the length of the fold? In other words, how would you choose x to minimize y?

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Homework Equations



None so far.

The Attempt at a Solution



I haven't a clue how to get started. I know I need to relate the length of y to the length of x and then differentiate and find the minima, but I don't know how to form the initial relationship.
 
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  • #2
Hint -- use the 8x12" information as part of the equations your write. If x is as shown, then the bottom piece is 8-x, right? What can you say about the angles in the two triangles that are formed by that fold?
 

1. What is an optimization problem?

An optimization problem is a mathematical problem that involves finding the best possible solution from a set of possible solutions. It involves maximizing or minimizing a certain quantity or objective function while satisfying a set of constraints.

2. How do you fold a triangle to minimize one side?

To fold a triangle to minimize one side, you need to use the geometric principle of reflection. First, draw a line from the vertex of the triangle to the midpoint of the opposite side. Then, fold the triangle along this line so that the two sides are reflected onto each other. This will result in the minimized side being the line of reflection.

3. What is the objective function in this optimization problem?

The objective function in this optimization problem is the length of the minimized side of the triangle. The goal is to minimize this length while satisfying the constraint of folding the triangle.

4. What are the constraints in this optimization problem?

The main constraint in this optimization problem is that the triangle must be folded in a way that results in a minimized side. Other constraints may include the length of the original sides of the triangle and the angle between the sides.

5. Can this optimization problem be solved using calculus?

Yes, this optimization problem can be solved using calculus. The process involves finding the derivative of the objective function and setting it equal to zero to find the critical points. Then, using the second derivative test, the optimal solution can be determined.

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