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## Meeting a Friend

Adapted from New South Wales Higher School Certificate 2005 Mathematics examination, Question 10 (copyright held by NSW Board of Studies).

Two friends agree to meet during their lunch hour, but both are very busy and unsure whether they can make it.

They each agree to wait for "x" minutes and, if the other has not arrived, to leave.

What is their chance of meeting? Drag the WaitTime point to explore this problem.

Let the unit square describe their lunch hour, and each point (x, y) within that square represent each of our times of arriving.

Then the point (1/2, 2/3) would indicate that I arrived at 12:30, and my friend arrived at 12:40.

How then do the inequalities x - y <= t and y - x <=t describe our chance of meeting?

What is the algebraic model that best fits this situation?

• 1. Drag the WaitTime point to find the probability of meeting within a 15 minute wait time.

• 2. Now find the length of the wait time required for a probability of 80%?

• 3(a) To build an algebraic model for the chance of meeting, observe that the action all occurs within a unit square, and the WaitTime point has coordinates (x, 0). Now begin by defining the function for the area of the triangle in the bottom right corner of the unit square:

tri(x) =

• 3(b) Notice that there is an identical triangle in the top left corner, together forming a square. The area we require for the probability of meeting is the difference between the area of the unit square and the total area of the two triangles. Try now to give the function meet(x) that describes this area/probability:

meet(x) =

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