4. Is it possible to predict the population of Georgia in the year 2025? Either explain why it cannot be done, or give a method for doing so.

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1 Homework Due Friday, Sept. 25 Population Explosion Homework 1. Notice that the population of Georgia rises dramatically in the second half of the 20 th century. Let us compute a new value of k by averaging the five intermediate k-values in your table from Use this new k to make a new prediction of the population of Georgia in Is this a better prediction than million? How do you know? 2. If the more recent data is more relevant, then the newest data should be the best. Compute the slope between the data points for the years 2000 and 2010, and let this be the value of P so that the value of k can be found. Make another prediction of the population in Is this a better prediction? Justify your answer. 3. The Government of the State of Georgia projects a population in Georgia of million in 2020 and million in Experiment with your calculator to determine a value of k that could arrive at these predictions. 4. Is it possible to predict the population of Georgia in the year 2025? Either explain why it cannot be done, or give a method for doing so. 5. Below is the population data for Gwinnett County, Georgia. Find an appropriate value of k for Malthus equation that models the growth of Gwinnett County. Is the value of k or Gwinnett County larger or smaller than the value of k for the state of Georgia? What does this imply about the growth of Gwinnett County? Predict the population of Gwinnett County in the year Population of Gwinnett 6. Either by hand or with a calculator, plot the population data for Gwinnett County County shown to the right. What curve does the data resemble? Malthus equation Year Population is more commonly referred to as the exponential growth equation does this (thousands) make sense based on the data? Suppose is a line. If the slope of is 3 and, then find Simplify Completely factor and Solve

2 Homework Due Friday, Sept. 25 Constrained Growth Homework 1. What is the average rate of change of the yeast population in Carlson s data over the first 17 hours? Indicate units of measure. 2. Using the sample bacterial growth we went over in class, Where when time, complete a table showing predicted population values for to hours. 3. Carlson s data for the yeast population can be modeled by the initial value problem Use the model to predict the yeast population for hours 1, 2, 3, and 4. Do the approximations given by the equation become better or worse than the actual values? Why do you think this is? 4. Determine the carrying capacity of the following logistic rate equations. a. b. c. d. 5. Consider the initial value problem This rate equation describes the rate of bombings in the United States since 1989, according to information released by the Bureau of Alcohol, Tobacco, Firearms, and Explosives (ATF). a. Make a table to record the predicted number of bombings for 1990 through b. The ATF specified that in 1994, a total of 3163 bombings occurred. How accurate is your predicted value for the number of bombings in 1994? c. According to the model, what should happen to the number of bombings in the future? d. What is your opinion of the validity of using this model to predict bombings in the next 10 years? 6. Suppose is a line. If the slope of is and, then find 7. A line with slope and passing through (2, 4) is parallel to another line passing through (-3, 6). Find the equations of both lines. 8. Completely factor and 9. Solve 10. Solve the system of equations {

3 Homework Due Friday, October 2 nd Newton s Second Law of Motion Homework 1. The barrels of nuclear waste can be safely dropped into the ocean wherever the ocen floor is less than how many feet deep? 2. What if the nuclear waste were housed in smaller barrels? If 40-gallon barrels were used, then and. Assume the drag constant remains Compute a velocity and distance table to determine whether the smaller barrels would be safer. 3. An object falling through air has a drag that is proportional to the square of the velocity. In other words,. Experiments show that the constant c for the 55-gallon barrel of nuclear waste is Determine the maximum distance the 55-gallong barrel could safely be dropped through air. (We can neglect the buoyancy b. Why?) 4. Consider the table for the 55-gallon barrel. Calculate left and right Riemann sums for the data set. How is this related to the distance table? 5. Fine the equation of the line that has x-intercept at 4 and y-intercept at A line with slope -3 and passing through (1, 5) is perpendicular to another line passing through (1, 1). Find the equations of both lines. 7. Completely factor and 8. Solve and 9. Solve the system of equations { 10. Find all the real zeros of 11. Completely factor

4 Euler s Method Homework Due Friday, October 2 nd 1. Use Euler s method on the initial value problem With to approximate. Do your answers make sense? Explain. 2. Use Euler s method on the initial value problem With to approximate. Show your work. 3. The velocity of an object is. Approximate the distance the object traveled in 2 seconds if the object was at position zero initially. 4. Given that the population of Georgia is million in the year 2010, use Euler s method on the initial value problem With to estimate the population of Georgia in the year How does this compare to the value you found on the Population Explosion homework assignment? 5. The rate at which a new chemical substance is formed from two chemicals is proportional to the amount, in moles, of the two chemicals present. In other words, if there are n moles of chemical N and h moles of chemical H, then the new substance, A, satisfies the rate equation. A mole of ammonia is made from one mole of nitrogen and three moles of hydrogen. If we begin with 20 moles each, then the rate is Where A is measured in moles and time is measured in seconds. Use Euler s method with number of moles of ammonia produced in 10 seconds. to estimate the 6. The Mauna Loa Observatory in Hawaii has a record of carbon dioxide levels in the atmosphere going back to March Levels of carbon dioxide are measured in parts per million (PPM). The average rate of change of CO 2 levels is PPM per year. a. Given an initial condition of PPM in March 1958, find the constant k so that the CO 2 levels can be modeled by a rate equation of the form, where C is measured in PPM. b. Use Euler s method on the rate equation in part (a) to approximate the CO 2 levels in c. The Intergovernmental Panel on Climate Change projects that CO 2 levels could reach PPM by 2050, possibly resulting in higher temperatures and rising sea levels. Does your mathematical model confirm this? Explain. 7. Consider the differential equation. Let be a particular solution to this differential equation with initial condition. Use Euler s method with two steps of equal size, starting at, to approximate. Show the work that leads to your answer. 8. Given a. Use Euler s method with two steps starting at to approximate given the initial condition b. Use Euler s method with two steps starting at to approximate given the initial condition

5 Slope Fields Homework Due Friday, October 2 nd 1. Plot, by hand, the slope field for for x:[0, 2] and y:[0, 2] 2. Plot the slope field for for x:[0, 2] and y:[0, 2]. Make a conjecture concerning the equation of the solution curve through the origin. Separate and integrate to find the actual solution don t forget C. 3. On each of the slope fields in the previous two problems, sketch, on each, a solution curve that passes through the point (1, 1). 4. Consider the rate equation Sketch the slope field for x:[0, 2] and y:[0, 2]. Separate and integrate to find the actual solution that passes through (- 1, 1) 5. Consider the population initial value problem a. Use Euler s method with to approximate. b. Generate a slope field for this rate equation. c. Separate and integrate to find the actual solution. d. Use that solution to estimate 6. Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears. Assuming a logistic growth model, when will the bear population reach 50? 75? 100? Remember, logistic growth model looked like: