Final: Bouncing Barney and Friends

by Laura Trkovsky

A. Bouncing Barney is an animation of Barney walking in a triangular room. He walks parallel to one side from another, and continues this until he ends up at his original starting point on the original wall. Below is an animation of this in gsp. Once in the GSP program click on the point to animate.

We can notice that the ball hits each side twice before it comes back to the same point it started from. We see this trend continues even when we move the starting point outside of the triangle because the bouncy ball will still not go out of the interior.

B. Find as many solutions as possible to:

It ends up being that you can pick the any of the two values, for instance when you solve the first equation for A you get:

This means that A depends on B and C. Thus B depends on A and C or C depends on A and B. Since this is the case there are multiple answers to this equation. To find solutions to this lets look at a graph in graphing calculator.

In this we can see that when C is greater than 4, then A and B have to be both positive. Then when C is between 0 and 4 there are no real solutions to this problem; however, when C is less than 0 either A or B has to be positive because of the multiplication going on. Therefore, there really are many solutions to this set of equations there are just certain limitations to what and when there are solutions.

C. Use excel to see the trends in the raising price of stamps. First we got the data from an already saved data file and added the 2006 increase to 39 cents. Next I went and made a chart using this data, and did all the specifications for the chart that I wanted. Next, I went in and added the trend lines of three different regression lines. I did a linear model, a exponential model, and a power model. Of these three the exponential model looked to be the line of best fit. So from there since I had the model. I could find out from these equations what year a particular amount of money will be required for a first class stamp. Here is the data and the chart with all the lines: EXCEL

Now let's explore when these values reach \$1.00 and 74 cents.

what we have to do is set the equation we choose to be best fit to these values. We get it will be 74 cents in 2020 and \$1.00 in the year 2027. The data has changed for many reasons because of inflation in the U.S. dollar. The model could have been true at the period in time, but since the cost of living and everything else has gone up so high this could cause the price to score as well. So obviously the trend has changed.

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