# Optimization of a folded piece of paper

• doug1122
In summary, the problem is to find the value of x that maximizes the area of a right triangle formed by folding one corner of an 8.5in by 11in piece of paper. The equation for the area of the triangle is A = 1/2(xy). Using the equation x2+y2=(8.5-y)2 and taking the derivative, we can find the maximum area by setting da/dx=0. This gives us the simplified equation x^2+y^2=(c-y)^2, where c is a constant.
doug1122

## Homework Statement

If you take an 8.5in by 11in piece of paper and fold one corner over so it just touches the opposite edge as seen in figure (http://wearpete.com/myprob.jpg ). Find the value of x that makes the area of the right triangle A a maximum?

A = 1/2(xy)
x2+y2=(8.5-y)2

## The Attempt at a Solution

x2+y2=(8.5-y)2
x = sqrt((8.5-y)2-y2)
A = 1/2(y)(sqrt((8.5-y)2-y2))
da/dx = ((y2-4.25y)/sqrt((8.5-y)2-y2))+1/2(sqrt(-17y-72.25))
I know that at da/dx=0 the triangle is maximized but da/dx is undefined at y=0 (y graphically). I am pretty sure my derivative is right but maybe I missed something there. Thanks for taking a look.

Last edited by a moderator:
at y = 0, the area will be zero, so i wouldn't be too concerned about that point

your method is ok, but could be simplified a bit... try muliplying out the RHS of your equation and simplifyng before substituting in
$$x^2+y^2=(8.5-y)^2$$

let 8.5 = c if it makes it easier
$$x^2+y^2=(c-y)^2$$

## 1. What is the purpose of optimizing a folded piece of paper?

The purpose of optimizing a folded piece of paper is to find the most efficient and effective way to manipulate the paper into a desired shape or structure, while minimizing wasted material and maximizing strength and stability.

## 2. What factors should be considered when optimizing a folded piece of paper?

Factors such as the type and thickness of the paper, the desired final shape or structure, and the intended use of the paper (e.g. for packaging, origami, etc.) should be considered when optimizing a folded piece of paper.

## 3. How can mathematical models be used in the optimization process?

Mathematical models can be used to predict and analyze the behavior of folded paper based on various factors, allowing for more precise and efficient optimization of the paper's shape and structure.

## 4. What are some common techniques for optimizing a folded piece of paper?

Some common techniques for optimizing a folded piece of paper include using symmetry and repetition, minimizing the number of folds, and incorporating geometric principles such as the golden ratio.

## 5. How does optimization of a folded piece of paper relate to real-world applications?

Optimization of a folded piece of paper has various real-world applications, such as in the design of efficient and sustainable packaging materials, the creation of complex and intricate origami artwork, and the development of foldable structures for space exploration and other industries.

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