March 25, 2017

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Linear Programming/Systems of Inequalities

The photocopying machine in a school office is made available to teachers between the hours of 3pm and 4pm. Mr. Grim and Mrs. Grump each have 10 minutes of copying to do each day. If theyh each enter the office at points in the available hour, what is the probability that one of them will have to wait while the other finishes copying?

Hi Sam. Welcome to Jiskha!

Draw yourself an x,y plot with both axes running from 0 to 50 minutes, representing the times that Grim (x) and Grump (y) might enter the room, measured from 3 PM. Assume those times are randomly distributed. The area of (x,y) space where |x-y| < 10 is the region where one or the other person is going to have to wait.

The probability that one of them will have to wait is the ratio or the area of a diagonal region between the lines x = y + 10 and x = y - 10, to the area of the 50 x 50 square. I get that ratio to be
1 - (2*0.5*40*40)/(50*50) = 1 - 16/25 = 9/25

The number (2*0.5*40*40)/(50*50) = 16/25 is the fraction of x,y space where no waiting is required, and is the ratio of the sum or two right-triangular areas to that of the 50 x 50 square

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