Message Board Assignment (Ch5) 4) Consider the graph to the right. Grassland butterflies - population index (1990 = 100) It comes from the article "The 120 European Butterfly Indicator for 100 Grassland Species: 1990-2007." The article discusses the causes of 80 this rapid decline, examining how 60 a decrease in biodiversity caused by human interactions with the 40 environment has affected 20 individual species of butterflies. a) A group of scientists is interesting in developing a 990 1992 1994 199 1998 2000 2002 06 200 20 mathematical model to better understand the behavior of the grassland butterflies population. Create a list of things that the scientists might want to consider in developing their mathematical model. b) Suppose one scientist wishes to model this population using a purely exponential model: E(t) = a . b', where E(t) represents the population index t years after 1990. Use the data points E(0) = 100 and E (10) = 70 to find values for a and b in our exponential model E (t). c) Suppose another scientist wishes to improve the model, choosing to use a model of the form P(t) = E(t) + V(t), where E (t) is the exponential function from part b, and V(t) represents the periodic variance, given by the function: V(t) = =1.05t - 1 In (1.05)/ sin(1.05t - t) + 0.05t2 Use the average value formula to find the average population index P(t) from the year 2000 to the year 2010. Round your answer to the nearest thousandth