Metropolitan Police Linear Problem

Topics: Rectangle, Automobile, Vertical direction Pages: 2 (575 words) Published: November 11, 2012
The Central Police Department had recently been criticized in the local media for not responding to police calls in the downtown area rapidly enough. In several recent cases, alarms had sounded for break-ins, but by the time the police car arrived, the perpetrators had left, and in one instance a store owner had been shot. Sergeant Davies had been assigned by the chief to find a way to determine the optimal patrol area (dimensions) for their cars that would minimize the average time it took to respond to a call in the downtown area. Sergeant Davies solicited help from Ms. Maris, an analyst in the operations area for the police department. Together they began to work through the problem. Sergeant Davies noted to Ms. Maris that normal patrol sectors are laid out in rectangles, with each rectangle including a number of city blocks. For illustrative purposes he defined the dimensions of the sector as x in the horizontal direction and as y in the vertical direction. He explained to Ms. Maris that cars traveled in straight lines either horizontally or vertically and turned at right angles. Travel in a horizontal direction must be accompanied by travel in a vertical direction, and the total distance traveled is the sum of the horizontal and vertical segments. He further noted that past research on police patrolling in urban areas had shown that the average distance traveled by a patrol car responding to a call in either direction was one third of the dimensions of the sector, or x/3 and y/3. He also explained that the travel time it took to respond to a call (assuming that a car left immediately upon receiving the call) is simply the average distance traveled divided by the average travel speed. Ms. Maris told Sergeant Davies that now that she understood how average travel time to a call was determined, she could see that it was closely related to the size of the patrol area. She asked Sergeant Davies if there were any restrictions on the size of the area sectors that...
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