Dutch elm disease is a fungal disease that kills elm trees. It is spread by certain species of bark beetle. The disease was first reported in the United States in 1928, with the
beetles believed to have arrived in a shipment of logs from the Netherlands destined for use as veneer in the Ohio furniture industry. The disease spread slowly from New England westward and
southward, almost completely destroying the famous elms in the “Elm City” of New Haven, reaching the Detroit area in 1950, the Chicago area by 1960, and Minneapolis by 1970. (Source: Wikipedia)
Ulysses Ulmus, a manager with the Maryland Department of the Environment, contacts you for help. He tells you about a region that had 8000 healthy elm trees in 2010. In that year, bark beetles
harboring the fungus arrived in the region. In each subsequent year, the number of elm trees that contracted the disease was 20% of the number of healthy trees at the end of the previous year. Once
a tree is infected, it stays infected. He asks you to do an analysis of the situation, addressing the following tasks: (a) Find a difference equation describing the spread of Dutch elm disease
through the region’s 8000 elm trees. Clearly define each variable. (b) Use your difference equation to find the total number of infected elm trees for each of the first six years after the arrival
of the beetles. Show all the details of every calculation. (c) Make a large, well-labeled graph showing the spread of Dutch elm disease, with total number of infected elm trees on the vertical axis
and time in years on the horizontal axis. Number each axis appropriately. Include 2010 and at least the next six years. The graph must be like the difference-equation graphs in the textbook and in
my solutions. I recommend that you use computer software to make the graph. If you do the graph by hand, you may either scan it and e-mail it to me, or bring it in to me on campus. If you do it by
hand, you must use graph paper and write your name on it. (d) Solve the difference equation. (e) Use the solved difference equation to find the number of infected elm trees at the end of 2030. (f)
Use the solved difference equation to find in what calendar year (like 2037) the number of infected elm trees will be 7560. You may solve the equation any way you like; you must explain what you
did Compose a letter to Mr. Ulmus, giving him the results of your analysis and explaining how you did it. You may use letters (a) through (I), as above, to label the parts of your solution.