Meeting a Friend
Author: Steve Arnold
Date: 06/09/2006

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 "t" minutes and, if the other has not arrived, to leave.

What is their chance of meeting?

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?

Wait time (minutes)

Chance of meeting

%

Chance (%)

(y)

(x)

Wait Time (hours)

O = (0, 0)
A = (0, 1)
B = (1, 1)
C = (1, 0)
P, o, a, b, c = Polygon(O, A, B, C)
d = Segment(O, C)
WaitTime = Point(d)
f = Distance(O, WaitTime)
g: y = x - f
e: y = x + f
E = Intersect(e, o)
F = Intersect(a, e)
G = Intersect(b, g)
Q, o_1, e_1, f_1, b_1, g_1, d_1 = Polygon(O, E, F, B, G, WaitTime)
h = Area(O, E, F, B, G, C, WaitTime, O)
W: (f, Q)
i = Segment(W, WaitTime)
R: (0, Q)
j = Segment(R, W)
Wait: 60 f
T1: Wait + ""
T4: 100 Q + ""
meet(x) = 0