7. Students who completed a class participated in a year-long study to see how much content...

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7. Students who completed a class participated in a year-long study to see how much content from the class they retained over the following year. At the end of the class, students completed an initial test to determine the group's content knowledge. At that time (t = 0), the group of students achieved a score of 75 out of 100 points. For the next 12 months, the group was evaluated at the end of each month to track their retention of the content. After 3 months (t = 3), the group's score was 70.84 points. The group's score can be modeled by the function R given by R(t) = a + b ln(t + 1), where R(t) is the score, in points, for month t, and t is the number of months since the initial test. (A) (1) Use the given data to write two equations that can be used to find the values for constants a and b in the expression for R(t). (ii) Find the values for a and b. (B) (i) Use the given data to find the average rate of change of the scores, in points per month, from t = 0 to t = 3 months. Express your answer as a decimal approximation. Show the computations that lead to your answer. (C) (ii) Interpret the meaning of your answer from (i) in the context of the problem. (iii) Consider the average rates of change of R from t = 3 to t = p months, where p > 3. Are these average rates of change less than or greater than the average rate of change from t = 0 to t = 3 months found in (i)? Explain your reasoning. The leaders of the study decide to use model R to make predictions about the group's score beyond 12 months (1 year). For a given year, model R is an appropriate model if the group's predicted score at the end of the year is at least 1 point lower than the group's predicted score at the end of the previous year. Based on this information, for how many years is model R an appropriate model? Give a reason for your answer. (Note: The end of a year occurs every 12 months from the initial evaluation-- t= 12, t = 24, ...) 7. Students who completed a class participated in a year-long study to see how much content from the class they retained over the following year. At the end of the class, students completed an initial test to determine the group's content knowledge. At that time (t = 0), the group of students achieved a score of 75 out of 100 points. For the next 12 months, the group was evaluated at the end of each month to track their retention of the content. After 3 months (t = 3), the group's score was 70.84 points. The group's score can be modeled by the function R given by R(t) = a + b ln(t + 1), where R(t) is the score, in points, for month t, and t is the number of months since the initial test. (A) (1) Use the given data to write two equations that can be used to find the values for constants a and b in the expression for R(t). (ii) Find the values for a and b. (B) (i) Use the given data to find the average rate of change of the scores, in points per month, from t = 0 to t = 3 months. Express your answer as a decimal approximation. Show the computations that lead to your answer. (C) (ii) Interpret the meaning of your answer from (i) in the context of the problem. (iii) Consider the average rates of change of R from t = 3 to t = p months, where p > 3. Are these average rates of change less than or greater than the average rate of change from t = 0 to t = 3 months found in (i)? Explain your reasoning. The leaders of the study decide to use model R to make predictions about the group's score beyond 12 months (1 year). For a given year, model R is an appropriate model if the group's predicted score at the end of the year is at least 1 point lower than the group's predicted score at the end of the previous year. Based on this information, for how many years is model R an appropriate model? Give a reason for your answer. (Note: The end of a year occurs every 12 months from the initial evaluation-- t= 12, t = 24, ...)