Question

The percent of paper and paperboard that is recycled from municipal solid waste has grown exponentially The table lists the percentages for selected years. Use a graphing calculator to fit the data to an exponential function Use the function to find R in 2009 In what year will the percentage exceed 75 Year, x Percent of paper 1980, 21.8 26.9 35.3 39.8 40.1 40.6 50.3 48.1 49.5 1990, 10 1994, 14 1995, 15 Type the values from your calculator for a and b for the exponential function y ab" 1996, 16 1997, 17 1998, 18 1999 19 (Round to six decimal places as needed) 2000 20 495 What is the approximate percentage of paper that will be recycled in 2009 ound to the nearest integer as needed) Inwhat year will the percentage exceed 75%? Round to the nearest year as needed ) Enter your answer in each of the answer boxes

Answer 1

Given that R(x) = y = ab^x put x = 0 then R(0) = 20.6 = ab^0 20.6 = a (1) when x = 10 R(10) = 26.4 = ab^10 but from a = 20.3 \r\n 20.6 middot b^10 = 26.4 b^10 = 26.4/20.6 b^10 = 1.28155 b = 10 squareroot 1.28155 b = 1.025 Therefore a = 20.6 b = 1.025 (iii) R(x) = y = ab^x y = 20.6 (1.025)^x in 2009, x = 29, so substitute x = 29 R(x) = y = 20.6 (1.025)^29 R(n) = y = 42.15 IV R(x) = 75 % R(x) = 20.6 (1.025)^x = 75 (1.025)^x = 3.640 \r\n taking log on both sides x log(1.025) = log(3.640) x(0.010) = 0.561 x = 0.561/0.010 = 56.1 Therefore If x = 56, R(x) = 75% Therefore in 2036, R(n) = 75%