*How big is the problem? **Create a model for the amount of plastic that ends up in landfills in the United States. Predict the production rate of plastic waste over time and predict the amount of plastic waste present in landfills 10 years from today.*

When I was thinking about this question in the Moody’s Math Challenge, Markov Chain initially came into my mind. Markov Chain was first introduced to me in my Linear Algebra at Westtown School class last year as an application of a matrix in the real world. According to the Linear Algebra textbook, the Markov Chains are used as mathematical models of a wide variety of situations in biology, business, chemistry, engineering, physics, and elsewhere. In each case, the model is used to describe a transformation that is performed many times in the same way, where the outcome of each transformation depends only on the immediately preceding one.

The reason that we considered using Markov chain in our situation is that we had found data online that said that the annual recycling rate for plastic wastes is approximately 5%. Now thinking back, Markov chain might not be the best option since in the real cases each year’s recycling rates are constantly changing instead of remaining the same. It means that when we chose the Markov Chain, we were assuming that starting from 2012 to the next 10 or 100 years the recycling rate is remaining at 5% for the plastic wastes. And so in my new paper for Question 1, I will focus more on the growing and changing rate of recycling and focus on what data say.

What I am working on now is considering the factor of population growth. The concept of population has never occurred to me before. The amount of landfilled plastic waste will be largely related to the American population in ten years. Below is my data collection for the American populations:

Years | American Population (millions) |

2011 | 3166 |

2010 | 3093 |

2009 | 3068 |

2008 | 3041 |

2007 | 3012 |

2006 | 2984 |

2005 | 2955 |

2004 | 2928 |

2003 | 2901 |

2002 | 2876 |

2001 | 2850 |

2000 | 2822 |

1999 | 2790 |

1998 | 2759 |

1997 | 2726 |

1996 | 2694 |

1995 | 2663 |

1994 | 2631 |

1993 | 2599 |

1992 | 2565 |

1991 | 2530 |

1990 | 2496 |

1989 | 2468 |

1988 | 2445 |

1987 | 2423 |

1986 | 2401 |

1985 | 2379 |

1984 | 2358 |

1983 | 2338 |

1982 | 2317 |

1981 | 2295 |

1980 | 2265 |

1979 | 2251 |

1978 | 2226 |

1977 | 2202 |

1976 | 2180 |

1975 | 2160 |

1974 | 2139 |

1973 | 2119 |

1972 | 2099 |

It is well known that we use exponential equations as the population model. Then I could use a data-analyzing tool such as Excel to form an exponential equation for the American population, which will lead me to the American population in 10 years and 100 years (although I will have more to say about this).

Then I focused on the data for the amount of plastic waste landfilled. Below are my data:

Years | Amount of Plastic Waste landfilled (Thousand tons) |

2011 | 29190 |

2010 | 28490 |

2009 | 27710 |

2008 | 27930 |

2007 | 28640 |

2006 | 27450 |

2005 | 27260 |

2004 | |

2003 | 25260 |

2002 | |

2001 | 23990 |

2000 | 23370 |

1999 | 23360 |

1998 | 20350 |

1997 | 18700 |

1996 | 17990 |

In the short term, or in this case, ten-year-time, the data reveal a linear growth, and so I think that I would go with a linear function for the estimated amount in ten years. In 100 years, however, it would be a totally different story. People begin to emphasize more about the idea of ecological economy, which will provide incentives to find alternative ways for landfill. Technology breakthroughs, which would seem improbable in a ten-year-period, would absolutely happen in the next 100 years. Therefore, if I have to make a prediction for 100 years later with a number and a function, I will choose a logarithmic tendency at the beginning and then switch to a declining function.

Hopefully, this part of the problem will be done in a week or two.

Shuangcheng Du