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How to Unwrap a Word Problem

How to Unwrap a Word Problem

 

Blog  9-7-15 Unwrapping a word problem

 

Here’s a recent word problem that I encountered that required a process to unwrap and solve:

 

At a 20 meter race Mike, Joe and Mary are competing to win. Mike can cover the distance in 5 seconds. Mary can run at a rate of 3 meters every 2 seconds and she thinks she can win if she starts three meters ahead. Joe begins at the start line but he can travel at 5 meters every two seconds.

Who will come in 1’st 2’ed and 3’ed. At what point will one of the remaining two runners begin to pass each other.

 

First we have to analyze rates of speed measured in meters per second. These are the slopes. On a graph the meters will be on the Y axes and the Seconds on the X axes. So if we look at the rise over run we get the rate in meters per second.

 

Mike can finish 20 meters in 5 seconds so his rate or slope is 20 meters/5 seconds which is 4 meters per. second.  His slope is 4 and being that he starts at the start line or (0,0) the equation for Mikes function is Y= 4X

 

Joe begins at the stare line and has a rate of speed of 5 meters every 2 seconds. So his slope is 5/2. Being that he starts at the start line or (0,0) the linear equation that describes his motion is Y= 5/2 X

 

Mary has a rate of 3 meters every 2 seconds so her slope is 3/2 but she begins at the 3 meter mark instead of at (0,0) so the equation that describes Mary’s movement is Y = 3/2 X + 3

 

If you plot these three equations on a graph you will see that Mike will clearly be the winner, Joe will be next and Mary will finish last even though she had a head start.

 

To find the point where Joe passes Mary you have to solve a system of equations setting Mary and Joe’s equations equal to each other:

5/2 X = 3/2 X + 3

X = 3  which means that they are tied at three seconds into the race

Joe will begin to pull ahead of Mary after three seconds.