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| == Explanation == | | == Explanation == |
− | This comic centers around the consideration of what is the shortest path available to a person traveling by foot. [[Cueball]] has to travel across a rectangular distance, which has an established path around the periphery. When Cueball follows these paths, he has to walk for 60 seconds. He realizes that by ignoring the paths and taking the {{w|Desire path|desire lines}} from corner to corner, his route will be shorter, and he calculates that he could cut up to 26% of his time. As a result, every time he has to travel this rectangle, he worries about the extra time taken as a result of following the path. There are downfalls to this plan, however. This is convenient for Cueball but probably not for the building owner, as many rectangular lawns have delicate decorations such as flowers on them. | + | This comic centers around the consideration what shortest path is available to a person travelling by foot. [[Cueball]] has to travel a rectangular distance, which allows only to walk over pavement. But the other paths over some other material not really meant to tread on (grass, sand or granite) are just shorter. When Cueball goes to follow the pavement, he has to walk for 60 seconds. But when he traverses using the restricted areas he can cut up to 26% of his time. So, every time he has to travel this rectangle he thinks about to shorten his way. Who would rather walk the straight path when an illegal way (to some standards) will save time? |
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− | Each path has labels for the time it takes (e.g. Path 2 takes 48.2 seconds) and the time compared to the longest path (e.g. Path 3 takes 74% as long as Path 1). Each path also has a corresponding equation for in the upper-right corner representing the time each path would take if Path 1 takes ''t'' seconds (instead of 60).
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− | ===Paths===
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− | Each path represents a different way of traveling on/through the two squares that make up the rectangle.
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− | Path 1 takes the long way around both squares. It takes 60 seconds in total, meaning it takes Cueball 20 seconds to walk across each of the three sides. By definition, it takes ''t'' seconds to walk the whole path and ''t''/3 seconds to walk each side.
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− | Path 2 takes the long way around one square but cuts diagonally across the other. Since each side takes 20 seconds, the total time is 20 (one side) plus 20√2 (the diagonal) seconds, which adds up to about 48.28 seconds. This is about 80.5% of the full, 60-second path. More generally, it takes t/3 + (t/3)√2, or t(1+√2)/3, seconds to walk the second path (though the percentage never changes).
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− | Path 3 cuts diagonally across the rectangle. The total time is the length of the diagonal, which is 20√5 (44.72...) seconds, per the Pythagorean theorem. This is about 74.5% of the the full path. Generally, it takes t√5/3 seconds to walk path 3. (As with path 2, and any other path that scales linearly with the full path length, the percentage doesn't change.)
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| ==Transcript== | | ==Transcript== |
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| {{comic discussion}} | | {{comic discussion}} |
− | [[Category:Geometry]] | + | [[Category:Math]] |
| [[Category:Comics featuring Cueball]] | | [[Category:Comics featuring Cueball]] |
| [[Category:Maps]] | | [[Category:Maps]] |
− | [[Category:Time management]]
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− | [[Category:Comics with lowercase text]]
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