A small country emits $54$ , $000$ kilotons of carbon dioxide per year. In a recent global agreement, the country agreed to cut its carbon emissions by $4 \%$ per year for the next $14$ years. In the first year of the agreement, the country will keep its emissions at $54$ , $000$ kilotons and the emissions will decrease $4 \%$ in each successive year. How many total kilotons of carbon dioxide would the country emit over the course of the $14$ year period, to the nearest whole number? $\newline$ Answer:

View step-by-step help

Home

Math Problems

Precalculus

Exponential growth and decay: word problems

Full solution

Q. A small country emits

54

000

kilotons of carbon dioxide per year. In a recent global agreement, the country agreed to cut its carbon emissions by

4 \%

per year for the next

14

years. In the first year of the agreement, the country will keep its emissions at

54

000

kilotons and the emissions will decrease

4 \%

in each successive year. How many total kilotons of carbon dioxide would the country emit over the course of the

14

year period, to the nearest whole number?

\newline

Answer:

Understand the problem: Understand the problem. $\newline$ We need to calculate the total carbon emissions over $14$ years, with a $4\%$ reduction each year after the first year. The initial emission is $54,000$ kilotons.
Calculate emissions for each year: Calculate the emissions for each year. $\newline$ The emissions for the first year are $54,000$ kilotons. For each subsequent year, the emissions are $4\%$ less than the previous year.
Use sum formula for series: Use the formula for the sum of a decreasing geometric series. $\newline$ The sum $S$ of $n$ terms of a geometric series with first term $a$ and common ratio $r$ is given by $S = \frac{a(1 - r^n)}{(1 - r)}$ , where $0 < r < 1$ . $\newline$ In this case, $a = 54,000$ kilotons, $r = 0.96$ (since there's a $4$ % reduction, $100\% - 4\% = 96\% = 0.96$ ), and $n = 14$ .
Calculate total emissions: Calculate the total emissions. $\newline$ $S = 54,000 \times (1 - 0.96^{14}) / (1 - 0.96)$ $\newline$ First, calculate $0.96^{14}$ .
Calculate $0.96^{14}$ : Calculate $0.96^{14}$ . $\newline$ $0.96^{14} \approx 0.563$
Substitute into sum formula: Substitute $0.96^{14}$ into the sum formula. $\newline$ $S = 54,000 \times (1 - 0.563) / (1 - 0.96)$
Calculate numerator of sum: Calculate the numerator of the sum formula. $\newline$ $1 - 0.563 = 0.437$ $\newline$ $S = 54,000 \times 0.437 / (1 - 0.96)$
Calculate denominator of sum: Calculate the denominator of the sum formula. $\newline$ $1 - 0.96 = 0.04$ $\newline$ $S = 54,000 \times 0.437 / 0.04$
Calculate total emissions: Calculate the total emissions $S$ . $\newline$ $S = 54,000 \times 0.437 / 0.04$ $\newline$ $S = 54,000 \times 10.925$ $\newline$ $S \approx 590,550 \text{ kilotons}$
Round total emissions: Round the total emissions to the nearest whole number. $S \approx 590,550$ kilotons