Meeting a Friend

©2020 Compass Learning Technologies ← Live Mathematics on the Web → Meeting a Friend
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?
Points which lie within the shaded hexagon correspond to times of arrival for which we would meet. Points outside the polygon are times for which we would not meet. In fact, our chance of meeting for any wait time, x, will correspond to the AREA of the shaded figure - and the moving point on the screen has coordinates (x, area(x)).
What is the algebraic model that best fits this situation? More simply, what is the formula for the area of the hexagon for any wait time, x?

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Assessment

Hint: When entering mathematical expressions in the math boxes below, use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.

Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". More?

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

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

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3. 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) =

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4. 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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Comments? Suggestions? What did you learn from this activity?

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This problem may also be modelled using CabriJr on the TI-83/84 series calculator. Click on the graphic below to download the file and try it yourself!

Try having students work in pairs: one generating data points using the CabriJr model and the partner entering these into lists to plot and then to help build their algebraic model!

Just wait until you try this with TI-nspire CAS!

And you really should have a look at GXWeb and Geometry Expressions.

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This dynamic figure was created using the latest version of the free GXWeb from Saltire Software, as shown below.

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©2019 Compass Learning Technologies ← Live Mathematics on the Web ← GeoGebra Showcase