This problem explores the world population growth. You can find a tremendous amount of material on this subject at the UN website. Below is a table from data during the 20^th century at this website on the world's population, where the numbers in the table are billions of people

a. We are beginning our studies of differential equations with the simplest mathematical model, the Malthusian growth model. Let P(t) be the population in billions, then a Malthusian growth model is given by the differential equation:

Use Maple's dsolve routine to find the general solution to this differential equation and paste the MAPLE inputs and outputs into your lab report.

b. Have Excel graph the data above. The solution in Part a. is an exponential function, so use Excel's Trendline (Exponential) to find the best fit to these data. What are the constants P₀ and r? (Be sure to have at least 3 significant figures for these constants.)

c. Exponentially growing populations are often plotted using semilog graphs. Create a graph in Excel with the vertical axis being logarithmic. What is the geometric shape of the Trendline in this graph?

d. Make a table of the percent error between the model and each of the data points in the table above. Use your mathematical model to predict the population in the years 1950, 2020, and 2050. Which of these predictions do you believe to be the best and why?

e. It has been suggested Earth can feed at most 12 billion people. In what year will the World reach this maximum carrying capacity according to your model.