5. Greenhouse gas emissions in the United States follow the exponential model: y

Question

5. Greenhouse gas emissions in the United States follow the exponential model: y =ye Suppose emissions are decreasing at the rate of 6% per year. Solve for the decay constant 'k . (Note: k Round 'k'correctly to 5 decimal places. a. 06 .) b. Assume we do not know our current emissions. How many years will it take for our emissions to drop to 10% of our current value? Let January 1, 2013 be time t -0. If our emissions in calendar year 2013 were 1800 megatons, how much greenhouse gas will we emit in the six years from January 1, 2016 through January 1, 2022? c.

Explanation / Answer

yt = y0 ekt
a) It is given that emissions are decreasing at the rate of 6% per year.
So y1 = y0 ek*1
But y1 = 0.94 * y0 (as emissions decreased by 6% in a year)
0.94 y0 = y0 ek
ek = 0.94 so
k = ln (0.94) =-0.0618754 ~= -0.06188
Thus, value of 'k' upto 2 decimal places is = - 0.06188

b) Let emissions drop to 10% of the current value after 't' years;
yt = y0 ekt
yt = 0.1y0
0.1y0 = y0 ekt
ekt = 0.1
kt = ln (0.1)
t= ln (0.1)/ k = -2.30259/ (-0.06188) = 37.21 years
Thus, the emission will drop to 10% of the current value after 37.21 years

c) Emissions in the calendar year 2013 = 1800 megatons;
so emissions from Jan 1 2016 to Jan 1 2022 are to be found out:
Emissions in the calendar year 2014 = 0.94 (1800)
Emissions in the calendar year 2015 = 0.942 (1800)
Emissions in the calendar year 2016 = 0.943 (1800)
Emissions in the calendar year 2017 = 0.944 (1800)
Emissions in the calendar year 2018 = 0.945 (1800)
Emissions in the calendar year 2019 = 0.946 (1800)
Emissions in the calendar year 2020 = 0.947 (1800)
Emissions in the calendar year 2021 = 0.948 (1800)

So for the period Jan 1 2016 to Jan 1 2022, add the emissions for the year 2016,'17,'18,'19,'20,'21
We get : 0.943 (1800) + 0.944 (1800) + 0.945 (1800)+ 0.946 (1800)+ 0.947 (1800)+ 0.948 (1800)
= 0.943 (1800) [ 1 + 0.94 + 0.942 + 0.943 + 0.944 + 0.955]
= 0.830584 (1800) ( 1 (0.94)6 -1 ) / (0.94-1)
= 1495.0512 ( 5.1688) = 7727.67 megatons

Thus, emissions in the calendar year 2016 to 2021 were 7727.67 MegaTons

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