# Maximum protrusion over a folded piece of paper [closed]

The pages of a book are $x$ cm tall and $y$ cm wide. If a page is folded appropriately, a corner of the page can stick out above the top of the book. What is the maximum amount that a page can protrude above the top of the book without tearing the page or separating it from the binding?

## closed as off-topic by Matthew Conroy, John B, A.Γ., mrp, R_DNov 30 '16 at 13:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Matthew Conroy, John B, A.Γ., mrp, R_D
If this question can be reworded to fit the rules in the help center, please edit the question.

I'll change the notation since I prefer using $x$ and $y$ as coordinates in the plane.
Let's say the edges of the page are on the lines $x=0$, $x=a$, $y=0$, $y=b$ before folding, and you fold along a line $y = c x + d$ with positive slope that crosses the top and bottom edges of the page, so $c > 0$, $0 \le (b-d)/c \le a$ and $0 \le -d/c \le a$. $$(x,y) \to \left(\frac{1-c^2}{1+c^2} x + \frac{2c}{1+c^2} y - \frac{2cd}{1+c^2}, \frac{2c}{1+c^2} x + \frac{c^2-1}{1+c^2} y + \dfrac{2d}{1+c^2}\right)$$ We want to maximize $$f(c,d) = \frac{2c}{1+c^2} a + \frac{c^2-1}{1+c^2} b + \dfrac{2d}{1+c^2}$$ subject to $c > 0$, $0 \ge d \ge b-ac$. Since $\partial f/\partial d = 2/(1+c^2) > 0$, the maximum will occur on the upper boundary of the region in $(c,d)$ space, i.e. at $d=0$. Taking the derivative with respect to $c$, we find that the solution is
$$c = \dfrac{b + \sqrt{a^2+b^2}}{a},\ d=0$$
$$\dfrac{a^2}{b+\sqrt{a^2+b^2}}$$