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. Assuming 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. Assuming 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?

Set up equation: Let's denote the number of years from today as $y$ . We need to set up an equation that represents the situation where the population of Swift Creek equals twice the population of Canyon Falls after $y$ years. $\newline$ Canyon Falls population after $y$ years: $22500 - 740y$ $\newline$ Swift Creek population after $y$ years: $15200 + 1500y$ $\newline$ The equation we need to solve is: $15200 + 1500y = 2(22500 - 740y)$
Distribute and combine terms: Now, let's distribute the $2$ on the right side of the equation: $\newline$ $15200 + 1500y = 45000 - 1480y$
Solve for y: Next, we'll combine like terms by adding $1480y$ to both sides and subtracting $15200$ from both sides: $\newline$ $1500y + 1480y = 45000 - 15200$ $\newline$ $2980y = 29800$
Final population after $10$ years: Now, we'll solve for $y$ by dividing both sides of the equation by $2980$ : $\newline$ $y = \frac{29800}{2980}$ $\newline$ $y = 10$
Final population after $10$ years: Now, we'll solve for $y$ by dividing both sides of the equation by $2980$ : $\newline$ $y = \frac{29800}{2980}$ $\newline$ $y = 10$ We have found that after $10$ years, the population of Swift Creek will be twice the population of Canyon Falls.