Question

46. Carbon Dioxide The table gives the estimated global carbon dioxide (CO2) emissions from fossil-fuel burning, cement pro duction, and gas flaring over the last century. The CO, esti- mates are expressed in millions of metric tons. Source: Carbon Dioride Information Analysis Center. CO, Emissions Year (millions of metric tons) 1900 534 1910 819 1920 932 1930 1053 1940 1299 1950 1630 1960 2569 1970 4053 1980 5316 1990 6151 2000 6750 2008 8749 a. Plot the data, letting = 0 correspond to 1900. Do the emissions appear to grow linearly or exponentially? b. Find an exponential function in the form of (1) that fits these data at 1900 and 2008, where is the number of years since 1900 and 7) is the CO, emissions c. Approximate the average annual percentage increase in CO, emissions during this time period. d. Graph/) and estimate the first year when emissions will be at least double what they were in 2008

Answer 1

Solution :

a) we have to plot the data

year	co2
1900	534
1910	819
1920	932
1930	1053
1940	1299
1950	1630
1960	2569
1970	4053
1980	5316
1990	6151
2000	6750
2008	8749

co2 Co2emmision 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 CO2 1880 1900 1920 1940 1960 1980 2000 2020 Year

Now letting t=0 for 1900 we can see that the co2 emmision grows exponentially.

b) We have to find the exponential function to estimate the CO2 emission for next t

where t is number of years from 1900

t	co2
0	534
10	819
20	932
30	1053
40	1299
50	1630
60	2569
70	4053
80	5316
90	6151
100	6750
108	8749

1. plot the graph of data and select any point We can get trend equation using MS excel as

2.right click on the point and select add trend line

3 select exponential and click on r squared and on equation as well.

4. we get the plot below

Plot to estimate CO2 y = 532.5e0.0262 R = 0.980 Co2 emission 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 co2 - Expon. (co2) 0 20 40 60 80 100 120 t

Here we get the eqution f(t)= 532.5 exp (0.026 * t)

We can fit the exp equation manually as

let y= abt

log y = log a +t *log b

let v= log y , A= log a , B = log b , U = (t - mean t)/10

Thus

v= A + BU

t	co2	U	v	U^2	UV
0	534	-5.48333	2.727541	30.06691	-14.956009
10	819	-4.48333	2.913284	20.10025	-13.061213
20	932	-3.48333	2.969416	12.13359	-10.343456
30	1053	-2.48333	3.022428	6.166928	-7.505687
40	1299	-1.48333	3.113609	2.200268	-4.6185099
50	1630	-0.48333	3.212188	0.233608	-1.5525466
60	2569	0.51667	3.409764	0.266948	1.76172282
70	4053	1.51667	3.607777	2.300288	5.47180654
80	5316	2.51667	3.725585	6.333628	9.37606793
90	6151	3.51667	3.788946	12.36697	13.3244718
100	6750	4.51667	3.829304	20.40031	17.2957015
108	8749	5.31667	3.941958	28.26698	20.9580921

Soving the normal equations

nA+ B $\sum$ U = $\sum$ v

A$\sum$U + B $\sum$ U2 = $\sum$ UV we get ,

f(t)= 532.5 exp (0.026 * t)

c) We have to approximate average annual increase in the CO2 emission

To find the annual growth rate is

gri = (ending value/ begining value) -1

e.g. gr1 = (819 / 534) -1

= 0.53

Accordingly we will calculate the growth rate for all t as follows

t	co2	annual growth rate
0	534	0
10	819	0.533707865
20	932	0.137973138
30	1053	0.129828326
40	1299	0.233618234
50	1630	0.254811393
60	2569	0.57607362
70	4053	0.577656676
80	5316	0.311621021
90	6151	0.157072987
100	6750	0.097382539
108	8749	0.296148148

Now we have to get the average annual growth rate

hence average annual growth rate = (gr1 + gr2+ ............... grn) /n

= 0.3005358

i.e 30.05 % is the average annual increase in CO2 during the given time year 1900 to 2008.

d) The plot of f(t) is as

Plot to estimate CO2 y = 532.5e0.0262 R = 0.980 Co2 emission 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 co2 - Expon. (co2) 0 20 40 60 80 100 120 t

Now we have to estimate f(t) for the year where CO2 emission will be doubled

in 2008 it was 8749

double of it is 8749*2= 17498

y(t) = 17498 = 532.5 exp (0.026 * t)

0.026* t = ln (17498 / 532.5)

thus t= 134.31

That is approximately( 1900+ 134.31 )year = 2034 year

hence in year 2034 the CO2 emission will be double of that in 2008.