Greenhouse Gases: Carbon dioxide (CO2) is a greenhouse gas in the atmosphere that may raise average temperatures on Earth. The burning of fossil fuels could be responsible for the increased levels of carbon dioxide. If current trends continue, future concentrations of atmospheric carbon dioxide in parts per million (ppm) could reach the levels shown in the accompanying table. The CO2 concentration in the year 2000 was greater than it had been at any time in the previous 160,000 years.
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.
(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 2085
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