Question
A sheet of paper 4 inches wide by 8 inches high is folded so that the bottom right corner of the sheet lies on the left hand edge of the sheet no more than 4 inches above the bottom of the sheet. The paper is creased, forming a crease from the bottom to the right edge. Find the length L, of the crease and how to fold the paper so that L is a minimum.
Answer
Draw a diagram of the folded sheet. On your diagram, label the following points:
Point A - Bottom left corner of the sheet;
Point B - Point where the crease begins along the bottom edge;
Point C - Point where the bottom right corner meets the left edge;
Point D - Point where the crease ends along the right edge;
Point E - Point along the left edge on horizontal line intersecting Point D;
For Triangle A-B-C:
Label the distance from A to B as d;
Label the distance from A to C as h;
Label the distance from B to C as x;
For Triangle C-D-E:
The distance from D to E is 4;
Label the distance from C to E as a;
Label the distance from C to D as y;
The length of the crease is the distance from Point B to Point D. Label this distance as z.
Triangle A-B-C is similar to Triangle C-D-E (their angles are equal).
y/x = a/d
a = (y^2 - 16)^0.5
d = 4 - x
Substituting for a and d:
y/x = (y^2 - 16)^0.5 / (4 - x)
Through algebra, you can reduce this equation to:
x^2 + y^2 = (x)(y^2) / 2
Noting that x^2 + y^2 = z^2, you can express z^2 entirely as a function of x:
z^2 = (x^3)/(x - 2)
To find the minimum value of z, you find the value of x that minimizes z^2, which is found by differentiating the above equation with respect to x and setting the derivative equal to 0:
(3) (x^2) (x - 2) - (x^3) (1) = 0
(2) (x^3) = (6) (x^2)
x = 3
This is the value of x that minimizes z^2. Plugging x = 3 into the equation for z^2:
z^2 = (3^3)/(3 - 2) = 27
The minimum value of z is found by taking the square root of 27 = 5.196... inches
Since d = 4 - x,
d = 1 inch
You fold the paper so the crease starts 1 inch to the right of the bottom left corner.
http://vrd.askvrd.org/services/answerschema.xml
