Today, the population of Canyon Falls is $22$ , $500$ and the population of Swift Creek is $15$ , $200$ . The population of Canyon Falls is decreasing at the rate of $740$ people each year while the population of Swift Creek is increasing at the rate of $1$ , $500$ people each year. Assumin; these rates continue into the future, in how many years from today will the population of Swift Creek equal twice the population of Canyon Falls?

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Exponential growth and decay: word problems

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Q. Today, the population of Canyon Falls is

22

500

and the population of Swift Creek is

15

200

. The population of Canyon Falls is decreasing at the rate of

740

people each year while the population of Swift Creek is increasing at the rate of

1

500

people each year. Assumin; these rates continue into the future, in how many years from today will the population of Swift Creek equal twice the population of Canyon Falls?

Define Variables: Let's define the variables: $\newline$ Let $P_{\text{CF}}$ be the population of Canyon Falls. $\newline$ Let $P_{\text{SC}}$ be the population of Swift Creek. $\newline$ Let $t$ be the number of years from today. $\newline$ We are given: $\newline$ $P_{\text{CF}} = 22,500$ $\newline$ $P_{\text{SC}} = 15,200$ $\newline$ The rate of decrease for Canyon Falls is $740$ people per year. $\newline$ The rate of increase for Swift Creek is $1,500$ people per year. $\newline$ We want to find $t$ such that $P_{\text{SC}} = 2 \times P_{\text{CF}}$ . $\newline$ We can write two equations that describe the populations of Canyon Falls and Swift Creek after $t$ years: $\newline$ $P_{\text{SC}}$ $0$ (since the population is decreasing) $\newline$ $P_{\text{SC}}$ $1$ (since the population is increasing) $\newline$ We want to find $t$ when $P_{\text{SC}}$ $3$ .
Set Up Equation: Let's set up the equation based on the condition that the population of Swift Creek equals twice the population of Canyon Falls: $\newline$ $15,200 + 1,500t = 2 \times (22,500 - 740t)$ $\newline$ Now we will solve for $t$ .
Solve Equation: First, distribute the $2$ on the right side of the equation: $\newline$ $15,200 + 1,500t = 45,000 - 1,480t$ $\newline$ Now, let's combine like terms by moving all terms involving $t$ to one side and constants to the other side: $\newline$ $1,500t + 1,480t = 45,000 - 15,200$ $\newline$ This simplifies to: $\newline$ $2,980t = 29,800$ $\newline$ Now, divide both sides by $2,980$ to solve for $t$ : $\newline$ $t = \frac{29,800}{2,980}$
Find Value of t: Perform the division to find the value of $t$ : $t = 10$ So, it will take $10$ years for the population of Swift Creek to be twice the population of Canyon Falls, assuming the rates of increase and decrease remain constant.