Year 2000 2050 2100 2150 2200

CO2 (ppm) 364 467 600 769 987

(a) Let x be in years, where x=0 corresponds to 2000, x=1 to 2001, and so on. Find values for C and a so that f(x) = Ca^x models the data.

x = 0 corresponds to 2000.

Given when year = 2000, CO2 (ppm) = 364

Put f(x) = 364 and x = 0 in f(x) = Ca^x

364 = Ca^0

364 = C*1

So C = 364

Thus the function changes to f(x) = 364a^x

x=50 corresponds to 2050

Given when year = 2050, CO2 (ppm) = 467

Put f(x) = 467 and x = 50 in f(x) = 364a^x

467 = 364a^50

Divide each side by 364

467/364 = a^50 We can write it as

a^50 = 467/364

Take power (1/50) on each side

(a^50)^(1/50) = (467/364)^(1/50) a = 1.005 (rounded to three decimal places)

Thus, function will be f(x) = 364*1.005^x

(b) Graph f and the data in the same viewing rectangle.

[pic]

(c) Use f(x) to estimate graphically the year when the carbon dioxide concentration will be double the preindustrial level of 280 ppm.

We have to find the year when level will be 560 ppm. We can note from the graph that level will be 560 ppm around x = 85. x = 85 corresponds to the year 2085.

Answer: year