Using an exponential function and integration, it is found that over the entire 11 year period, the country will emit 763,968 kilotons of carbon dioxide.
Exponential function:
A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
In which:
- A(0) is the initial value.
- r is the decay rate, as a decimal.
In this problem:
- In the first year, the country emits 80,000 kilotons, hence [tex]A(0) = 80000[/tex].
- Emissions will decrease 2.6% in each successive year, hence [tex]r = 0.026[/tex]
Then, the equation for the amount emitted each year is:
[tex]A(t) = A(0)(1 - r)^t[/tex]
[tex]A(t) = 80000(1 - 0.026)^t[/tex]
[tex]A(t) = 80000(0.974)^t[/tex]
The total amount emitted over the course of the 11 year period is:
[tex]T = \int_{0}^{11} A(t) dt[/tex]
Hence:
[tex]T = \int_{0}^{11} 80000(0.974)^t dt[/tex]
[tex]T = \frac{80000(0.974)^t}{\ln{0.974}}|_{t = 0}^{t = 11}[/tex]
Applying the Fundamental Theorem of Calculus:
[tex]T = \frac{80000(0.974)^{11}}{\ln{0.974}} - \frac{80000(0.974)^0}{\ln{0.974}} = \frac{80000}{\ln{0.974}}((0.974)^{11} - 1) = 763968[/tex]
Over the entire 11 year period, the country will emit 763,968 kilotons of carbon dioxide.
You can learn more about exponential functions at https://brainly.com/question/25537936
