Problem 416

Sir SagarDecember 27, 2014December 28, 20140

Greenhouse Gases :- Carbon dioxide $(CO_{2})$ is a green-house gas in the atmosphere that may raise average temperatures on Earth. The burning of fossil fuels could be responsible for the increase 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 $(CO_{2})$ 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
$(CO_{2})$	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(a) = $Ca^{x}$ models the data.

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

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

Solution

A.

f(x) = $Ca^{x}$

Form table year = 2000 (x = 0) f(0) = 364

Now put both values in the given equation

364 = $C(a)^{0}$

364 = C

Now from take year = 2100 (x = 100), f(100) = 600, and C = 364

600 = $364 (a)^{1000}$

$(a)^{1000}$ = $\frac{600}{364}$

a = ( $\frac{600}{364})^{\frac{1}{100}}$

a = 1.005

So final value of C = 364 and a = 1.005

B.

Graph

C.

Form graph x is around 85 when $CO^{2}$ concentration is double of 280 ppm.

So year will be 2000 + 85 = 2085.

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