Stephen ARNOLD
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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).

### Learning Objectives

1. Model, apply and interpret relationships, including simple inequalities, involving variables related to a given context.
2. Interpret the variables involved in using some simple non-linear functions to model situations and make related predictions.
3. Use known formulas to develop, apply and interpret new relationships.

Two friends, Xavier and Yvette, agree to meet during their lunch hour (between 12 noon and 1 pm), but both are very busy and unsure whether they can make it.

They each agree to wait for a certain number of minutes and, if the other has not arrived, to leave.

Let the unit square describe their lunch hour, and each point (x, y) within that square represent each of their times of arriving. The point (1/2, 2/3) would indicate that Xavier arrived at 12:30, and that Yvette arrived at 12:40.

Drag the point labelled "WaitTime" on the figure below to explore their chance of meeting, and then answer the questions below.

 This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com 1. The friends agree to wait for 15 minutes. Xavier arrives first, at 12:15 pm. What are the coordinates of the point in the unit square corresponding to Xavier's arrival at 12:15 and Yvette arriving at the latest time possible for them to meet. [LO2] 2. Drag the point "WaitTime" to find their chance of meeting within that 15 minute wait time. [LO1] 3. Now drag "WaitTime" to find the length of the wait time required for a chance of 80%? [LO1]
• 4. If they agree to wait 15 minutes (1/4 hour) then explain the significance of the inequality x - y < 0.25 to your understanding of the question (you should include in your response reference to the corresponding inequality, y - x < 0.25). [LO2]

• 5. How might you find the area of the hexagon formed by the intersection of the inequalities and the unit square? Carefully explain the significance of this hexagon to this problem. [LO3]

• 6. Suppose the variable point labelled WaitTime lies x units from the origin. Build an algebraic model for the chance of meeting by defining the following function: [LO3]

meet(x) =

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• 7. [EXTENSION 1] Suppose Yvette is more patient than Xavier. While Xavier agrees to wait for 15 minutes (one quarter of an hour), his friend is willing to wait for 20 minutes (one third of an hour) before leaving. How does this change the geometric model we have created (think in terms of the square and inequalities that define our chance of meeting)? [LO3]

• 8. [EXTENSION 2] If Xavier am willing to wait for 15 minutes and Yvette for 20 minutes, what is their new chance of meeting? [LO3]

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