06/18/2020 (2011: National Target Round, Problem 4)

Q: The population of the town of Euler is growing at a rate of 3% per year. If the population is currently 35,000, after how many whole years will the population first exceed 50,000?

A: 13 years

Because the rate of growth is 3%, the current population is 35000 and we are trying to find out how many years it takes for the population to reach 50000, we can set up the equation: (35000)(1.03)^x = 50000, where x is the number of years.

We simplify that to: (1.03)^x = 10/7 --> ((1.03)^x)/(10/7) = 1 --> ((1.03)^x)/(10/7) - 1 = 0

We can use our calculators to graph it (as ((1.03)^x)/(10/7) - 1 = y) and we get: 12.067, which is greater than 12, so the answer is 13 years.