Problem of the Week #57: Monday August 5th, 2024
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by this twitter post.
If we draw a line through the center of a square, we can fold the square in half along this line. Depending on the angle this line makes with the horizontal, as shown in the diagram, the folded square will have different looks. For example, below are the square folded along the lines that are 0°, 45°, and 22.5° from the horizontal.
a). Find the area of the overlapping region when we fold along the line 22.5° from the horizontal.
b). With proof, find the angle of the folding line from the horizontal that would result in the smallest overlapping area.
c). Find the angle of the folding line from the horizontal and the area of the overlapping region in the rectangle shown.
d). Find the angle of the folding line that would result in the smallest overlapping area starting with a rectangle 1 unit wide and 2 units long.
e). Find the angle of the folding line that would result in the smallest overlapping area starting with a rectangle w units wide and l units long.
f). Sketch the graph of the overlapping area as a function of the angle of the folding line from the horizontal when folding a wxl rectangle through the center.
g). Find the angle of the folding line from the horizontal that would result in the smallest overlapping area when folding a regular n-gon.
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.