Problem 2.1 Finding the Point of Intersection

Today in class we learned how to find the variable so that the two equations are equal. This is the problem that we focused on.

In Ms. Chang's class, Emile found out that his walking rate is 2.5 meters per second. When he gets home from school, he times his little brother Henri as Henri walks 100 meters. He figured out that Henri's walking rate is 1 meter per second.
Henri challenges Emile to a walking race. Because Emile's walking rate is faster, Emile gives Henri a 45-meter head start. Emile knows his brother would enjoy winning the race, but he does not want to make the race so short that it is obvious his brother will win. How long should the race be so that Henri wins a close race?

One option is to use a graph. First, you plot the points to represent how far they have walked after every ten meters and connect them. Then you look to see where the two lines intersect to find the distance they will have to walk to meet. If you subtract a meter from that, Henri will only win by a little. The lines on the graph intersect at 75 meters so if you subtract one and make it 74, Henri will win a close race.

Another option is to use a table. This is what it would look like:

Time___Emile __Henri
_ 0 ____ 0______45
_10____ 25_____ 55
_20____ 50_____65
_30____ 75_____75
_40____100____ 85

(The time is in seconds and the data under Emile and Henri represent how far they have walked in meters.)

After 30 seconds they meet at 75 meters. If you make the race 74 meters, Henri will win but just by a little.

A third option to solve this is to use an equation. That is what I did.

2.5x = 1x + 45
2.5x - 1x = 1x + 45 - 1x
1.5x = 45
1.5x / 1.5 = 45 / 1.5
x = 30

2.5 = Emile's walking rate
x = seconds
1 = Henri's walking rate
45 = Henri's starting point

After 30 seconds they will meet. To figure out how far they will be from the starting point you have to do the check:

2.5(30) = 1(30)+45
75 = 30+45
75 = 75

After you do the check you figure out that at 75 meters they will meet. You need to subtract a meter so that Henri will win. After they have walked 74 meters, Henri will only be ahead by a little so he would win a close race.

From the graph, table and equation you can figure out that they will meet after 30 seconds at 75 meters. For Henri to win a close race, it would have to be 74 meters long.

Post by: Alex S.