Suppose a 20-sided polygon (example shown below) represents the floor plan of an art museum or...

 

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Suppose a 20-sided polygon (example shown below) represents the floor plan of an art museum or a bank. Guards or sensors are to be placed at some of the vertices so that if a "thief" were in the interior of the polygon, an alarm will sound. An interior point / is visible from point P on the boundary, where P is not necessarily a vertex, if the line segment from P to / contains no points in the exterior of the polygon. A polygon with a set of locations for the sensors is said to be well guarded if every point in its interior is visible from at least one sensor, and hence any "thief" in the interior will be detected. a. What is the smallest number of sensors located at the vertices needed for the 20- sided polygon shown to be well-guarded? b. The number of sensors you find in part a might not be enough for every 20-sided polygon. So then what is the largest number of sensors located at the vertices that might be required for SOME 20-sided polygons to be well-guarded? (You can consider convex or non-convex 20-sided polygons) c. Find a formula as a function of n for how many guards (sensors) located at the vertices of any n-sided polygons are needed to guarantee that a "thief" into the interior of the polygon will be detected. Suppose a 20-sided polygon (example shown below) represents the floor plan of an art museum or a bank. Guards or sensors are to be placed at some of the vertices so that if a "thief" were in the interior of the polygon, an alarm will sound. An interior point / is visible from point P on the boundary, where P is not necessarily a vertex, if the line segment from P to / contains no points in the exterior of the polygon. A polygon with a set of locations for the sensors is said to be well guarded if every point in its interior is visible from at least one sensor, and hence any "thief" in the interior will be detected. a. What is the smallest number of sensors located at the vertices needed for the 20- sided polygon shown to be well-guarded? b. The number of sensors you find in part a might not be enough for every 20-sided polygon. So then what is the largest number of sensors located at the vertices that might be required for SOME 20-sided polygons to be well-guarded? (You can consider convex or non-convex 20-sided polygons) c. Find a formula as a function of n for how many guards (sensors) located at the vertices of any n-sided polygons are needed to guarantee that a "thief" into the interior of the polygon will be detected.

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For every ata polygon guardd L Convex at and parts 86 conver ... View the full answer