25. Paper folding A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it with paper.

a. Show that L^2=2x^3/(2x-8.5).

Verified step by step guidance

1. Step 1: Begin by analyzing the geometry of the problem. The paper is folded such that one corner (A) touches the opposite longer edge (BC) at point Q. The crease forms a line segment L, and the goal is to express L^2 in terms of x, the distance from point P to point B along the bottom edge.
2. Step 2: Use the Pythagorean theorem to relate the dimensions of the triangle formed by the crease. The crease L is the hypotenuse of the triangle with one leg being x (distance AP) and the other leg being the vertical distance from R to the bottom edge (denoted as sqrt(L^2 - x^2)).
3. Step 3: Recognize that the vertical distance from R to the bottom edge is determined by the folding geometry. The paper's width is 8.5 inches, and the length is 11 inches. The folding creates a relationship between x and the vertical distance, which can be derived using similar triangles or geometric constraints.
4. Step 4: Derive the formula for L^2 using the relationship between x and the vertical distance. The crease length L depends on the geometry of the fold, and the problem specifies that L^2 = 2x^3 / (2x - 8.5). This formula can be derived by substituting the geometric relationships into the Pythagorean theorem and simplifying.
5. Step 5: Verify the formula by substituting values or analyzing the constraints. Ensure that the derived formula satisfies the geometric conditions of the problem and correctly represents the relationship between L^2 and x.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (L) is equal to the sum of the squares of the lengths of the other two sides. This theorem is essential for calculating distances in geometric problems, such as finding the length of the crease in the paper folding scenario.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we aim to minimize the length of the crease (L) by manipulating the position of the corner of the paper, which requires setting up a function and using techniques such as derivatives to find critical points.

Coordinate Geometry

Coordinate geometry uses algebraic equations to describe geometric figures in a coordinate system. In this problem, the positions of points on the paper can be represented using coordinates, allowing for the application of algebraic methods to derive relationships, such as the equation L^2 = 2x^3/(2x-8.5).