MONT 102N – Modeling The Environment Final Examination – December 14, 2011

Directions: Do all work on these sheets. You may use the back if necessary, but please alert me with a note if you do. In a case where a later part of a question depends on a previous part, I will give credit for the later part as long as you have a correct approach, even without the correct value from the previous part. There are 200 points, distributed as indicated in the questions.

I. In 1990, forests covered 4.047 × 10^{9} hectares of the Earth’s surface. By 2000, forest area had decreased to 4.038 × 10^{9} hectares. Assuming that the decrease in forest area is linear, and that it will continue at the same rate into the future, in this problem you will develop a linear model for the forest area remaining as a function of t = years since 1990.

A. (10) Determine the slope for the linear model of the forest area.

B. (5) What is the linear equation modeling the forest area as a function of t = years since 1990.

C. (10) Use your equation to predict the amount of forest area that will remain in 2010.

D. (10) According to your model, in what year will the forest area reach 4.0×10^{9} hectares?

E. (5) According to the United Nations Food and Agriculture Organization, the actual forest area remaining in 2010 was 4.033 × 10^{9} hectares. Discuss the relation between your prediction in part C and the actual figure. What is happening?

II. Wind power has emerged as the fastest growing source of energy for electrical power generation in recent years. In 2004, the generating power of wind turbines was about 40, 000 megawatts and it was increasing at about 25.99% per year.

A. (10) The typical English unit of power is the horsepower. 1 horsepower = .0007457 megawatts. Convert 40, 000 megawatts to the equivalent number of horsepower.

B. (10) Construct an exponential model for W P = wind power generation as a function of t = years since 2004. Use units of 10^{4} megawatts for WP.

C. (15) Fill in the table of values for W P below with values predicted by your model for the years 2004 − 2010. Round to 2 decimal places. In what year did W P reach approximately double the 2004 level?

Y ear 2004  2005  2006  2007  2008  2009  2010

W P

D. (10) How many years will it take for wind power generation to reach 320, 000 megawatts according to your model?

III. Suppose that a population of fast-reproducing insects in an area has a natural growth rate of 5% per month from births and deaths, and that there is a net migration loss of 20 individuals per month.

A. (5) Which of the following difference equation models for P (n) = population in month n fits the description above? (Place a check next to the correct one.)

1) P (n) = 5P (n − 1) + 20

2) P (n) = 1.05P (n − 20)

3) P (n) = 1.05P (n − 1) − 20

4) P (n) = 1.05P (n − 1) + 20

B. (15) Using an initial value P (0) = 500, determine the populations in months 1, 2, 3, 4, 5 according to the model you picked in part A and record the values in the following table (round any decimal values to the nearest whole number)

n  0  1  2  3  4  5
P (n) 500

C. (5) What happens to the population in the long run as n increase? Does it tend to a definite value?

IV. Answer any two of the following three questions (only the best two will be counted if you answer more than two).

A. (15) If you are fitting a power law model to a data set (x_{i}, y_{i}) “by hand,” you start by transforming the data to (X, Y) = (log(x_{i}), log(y_{i})). If the best fit regression line for the transformed data is Y = mX + b, what is the corresponding power law model? (Assume the logarithms have base 10 as we discussed in class.)

B. (15) What does the correlation coefficient r (or its square r^{2}) measure? How did we use it? Explain what it would mean, for instance if r = 1 or r = −1.

C. (15) What type of chart (scatterplot, pie chart, bar chart, etc.) would be most useful to describe the composition of a forest if there 5 different types of trees present in different concentrations per acre? Explain, and illustrate your answer with a chart if a typical acre of forest contains 10 oaks, 12 maples, 5 pines, 2 hemlocks, and 1 chestnut.

V. Essay. (60) What are the major reservoirs of carbon in the Earth’s short-term carbon cycle? What are the major flows (“fluxes”) of carbon between those reservoirs. Describe them in words and in a diagram. In our final project, we looked at a simple model of this system. What features of the real-world carbon cycle did that model include? What are three possibly important features of the real-world carbon cycle that were left out?