To find the carbon-dioxide emissions [tex]\( E(t) \)[/tex] [tex]\( t \)[/tex] years from today, we need to consider the following information:
- The current emission level is 40 million tons.
- Each year, the emission is reduced by 35%.
Let's denote:
- The initial emission level as [tex]\( E_0 \)[/tex] which is 40 million tons.
- The annual reduction rate as [tex]\( r \)[/tex] which is 35%, or 0.35 in decimal form.
We need to express [tex]\( E(t) \)[/tex], which is the emission level [tex]\( t \)[/tex] years from today. Each year, the emissions are reduced to 65% of the previous year's emissions. This means that we multiply the emission level from one year by 0.65 to get the next year's emission level (since [tex]\( 100\% - 35\% = 65\% \)[/tex]).
So, every year, the emissions will be multiplied by [tex]\( 0.65 \)[/tex]:
1. After 1 year, the emission level will be [tex]\( 40 \times 0.65 \)[/tex].
2. After 2 years, it will be [tex]\( 40 \times 0.65^2 \)[/tex].
3. After [tex]\( t \)[/tex] years, it will be [tex]\( 40 \times 0.65^t \)[/tex].
Therefore, to express the carbon-dioxide emissions [tex]\( E(t) \)[/tex] [tex]\( t \)[/tex] years from now, we use the formula:
[tex]\[
E(t) = 40 \times 0.65^t
\][/tex]
This is a function that describes Freedonia's carbon-dioxide emissions in million tons [tex]\( t \)[/tex] years from today.
