Squishing a Semi-arc
An arc is suspended between two lines at fixed points and on the line such that segment is perpendicular to both of the lines. The arc has length 100 units and segment has length 90 units. The two lines move closer and closer at a rate of 2 units per minute, thus squishing the arc. After how many minutes does arc no longer become an arc?
Details and assumptions
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Arc obeys physical properties, and bends in the most natural way possible
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Arc never goes through either of the lines suspending it
The answer is 13.169.
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1 solution
The "arc" will continue to be an arc until it no longer could possibly be an minor arc. In other words, after it's become a perfect semicircle, it will no longer be able to be fit onto a circle, which intuitively makes sense.
In nature, things bend like perfect curves because it is the path of least resistance. If you squish together the ends of a piece of paper lying flat on a table, the paper doesn't just fold up like a birthday card, it bends.
So when A B becomes exactly half of a circle, then 1 0 0 (the length of arc A B ) must be half the circumference. Thus, the diameter (which is also the length of segment A B must be π 2 0 0 . Since when it starts, segment A B is 90 units, it must travel 9 0 − π 2 0 0 . Thus at a rate of 2 units per minute, it takes about 13.169 minutes until it is no longer an arc.