The country of Freedonia has decided to reduce its carbondioxide emission by $35 \%$ each year. This year the country emitted $40$ million tons of carbon-dioxide. $\newline$ Write a function that gives Freedonia's carbon-dioxide emissions in million tons, $E(t), t$ years from today. $\newline$ $E(t)=\square$

View step-by-step help

Home

Math Problems

Algebra 2

Interpret the exponential function word problem

Full solution

Q. The country of Freedonia has decided to reduce its carbondioxide emission by

35 \%

each year. This year the country emitted

40

million tons of carbon-dioxide.

\newline

Write a function that gives Freedonia's carbon-dioxide emissions in million tons,

E(t), t

years from today.

\newline

E(t)=\square

Understand Exponential Decay: To create a function that models the carbon-dioxide emissions in Freedonia $t$ years from today, we need to understand that a reduction of $35\%$ each year means that each year, Freedonia will emit only $65\%$ ( $100\% - 35\%$ ) of the carbon-dioxide it emitted the previous year. This is an example of exponential decay, and the general formula for exponential decay is: $\newline$ $E(t) = E_0 \times (1 - r)^t$ $\newline$ where: $\newline$ $E(t)$ is the amount after $t$ years, $\newline$ $E_0$ is the initial amount (the amount today), $\newline$ $r$ is the decay rate (as a decimal), and $\newline$ $t$ is the time in years. $\newline$ Given that the initial amount of carbon-dioxide is $35\%$ $0$ million tons and the decay rate is $35\%$ , or $35\%$ $2$ as a decimal, we can substitute these values into the formula.
Convert Percentage to Decimal: First, we convert the percentage reduction to a decimal by dividing by $100$ . So, $35\%$ becomes $0.35$ . $\newline$ Next, we subtract this decimal from $1$ to find the percentage that remains each year. This gives us $1 - 0.35 = 0.65$ . $\newline$ Now we can write the function for $E(t)$ using the initial amount of carbon-dioxide, which is $40$ million tons, and the decay factor, which is $0.65$ . $\newline$ $E(t) = 40 \times (0.65)^t$
Write Emission Function: The function $E(t) = 40 \times (0.65)^t$ now represents the carbon-dioxide emissions in million tons, $t$ years from today. This function can be used to calculate the emissions for any given year $t$ by substituting the value of $t$ into the function. $\newline$ To check for any mathematical errors, let's consider the case when $t = 0$ , which should give us the initial value of $40$ million tons. $\newline$ $E(0) = 40 \times (0.65)^0 = 40 \times 1 = 40$ $\newline$ Since any number to the power of $0$ is $1$ , and $40$ times $1$ is $40$ , the function checks out for $t = 0$ .