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. Assuminj; 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?

View step-by-step help

Home

Math Problems

Precalculus

Exponential growth and decay: word problems

Full solution

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. Assuminj; 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 when the population of Swift Creek equals twice the population of Canyon Falls as ' $y$ '. We need to set up an equation that represents the populations of Canyon Falls and Swift Creek in ' $y$ ' years. $\newline$ Canyon Falls population in ' $y$ ' years: $22,500 - 740y$ $\newline$ Swift Creek population in ' $y$ ' years: $15,200 + 1,500y$ $\newline$ We want the population of Swift Creek to be twice that of Canyon Falls, so we set up the equation: $\newline$ $15,200 + 1,500y = 2(22,500 - 740y)$
Solve Equation: Now we will solve the equation for 'y'. $\newline$ $15,200 + 1,500y = 45,000 - 1,480y$ $\newline$ Combine like terms by adding $1,480y$ to both sides of the equation: $\newline$ $15,200 + 1,500y + 1,480y = 45,000$ $\newline$ $15,200 + 2,980y = 45,000$
Combine Like Terms: Next, we subtract $15,200$ from both sides to isolate the term with 'y': $\newline$ $2,980y = 45,000 - 15,200$ $\newline$ $2,980y = 29,800$
Isolate 'y' Term: Now, we divide both sides by $2,980$ to solve for 'y': $\newline$ $y = \frac{29,800}{2,980}$ $\newline$ $y = 10$