Be sure to show all work and all problem solving strategies. Give complete explanations for each step. To study how species survive, scientists model their populations by observing the dif- ferent stages in their life. Scientists consider, for example, the stage at which the animal is fertile, the proportion of the population that reproduces, and the proportion of the young that survive each year. In one study, scientists looked at an animal that is consid- ered immature for the first year of its life, a juvenile for the second year of life, and an adult for the remainder of its life. The following data was collected for this species: Immature Juvenile Adult 0 0 0.4 A= Immature Juvenile 0.1 0 0 600 Immature Xo 400 Juvenile 3500 Adult 0 0.3 0.8 Adult The entries in matrix A represent the percentage of the population that survives to the next year. For example, in the first column, 0% of the immature animals remain imma- ture, 10% become juveniles and, of course, none become adults (that takes two years). Of the juveniles, none go backwards to become immature and 30% become adults. For the adults, 40% reproduce and add to the immature population, none become juveniles, and 80% go on to the next year (still as adults). The matrix Xo represents the number of immature, juvenile, and adult animals in year 0. Let X₁ = AX0, X2= AX1, X3= AX2, etc. 1. Explain why X, gives the population in year 1, X₂ gives the population in year 2. and so on. 2. Find the population matrices for years 1, 2, 3, and 4. Round to the nearest whole numbers. Do you see any trends? 3. Show that X₂ = A²X0₁ X₁ = A³ Xo, etc. 4. Find the population after 50 years that is, find Xso (Use your result from #3 and a graphing calculator, don't try to do this by hand.) Does it appear that the population survives?