Math Help - Optimization: Minimize Paper

1. Optimization: Minimize Paper

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

2. Re: Optimization: Minimize Paper

Originally Posted by AXQ4286
The upper right-hand corner of a piece of paper, 12 in by 8 in, as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
I don't know what you have done so far and where you are stuck ... So here are some hints:

1. I've modified you sketch a little bi( see attachment)

2. The green-bordered triangle is congruent to the blue-bordered triangle.
3. x must be longer than 4.
4. The two colour-bordered triangles form a rhombus whose area is

$\frac12 \cdot y \cdot f = 2 \cdot \frac12 \cdot x \cdot k$

5. You are dealing with a lot of right triangles. For instance:

• $h^2+(8-x)^2=x^2$
• $h^2+8^2=f^2$
• $x^2+k^2=y^2$
• $(k-h)^2+8^2=k^2$

... and now it's your turn.

3. Re: Optimization: Minimize Paper

Originally Posted by AXQ4286
The upper right-hand corner of a piece of paper, 12 in by 8 in, as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
I've constructed some of all line segments y. The radius of the circles is x.

You can use this drawing to confirm the result of your calculations. (Btw: The minimum of y is not drawn!)