Let the population of a city be 𝑥. If the population increases by 13 percent each year, what will the population of the city be in nine years’ time?
The population increases by 13 percent each year, so the increase will get bigger every year. And we call this exponential growth. We have a general equation that we use to model exponential growth. 𝑦 equals 𝑎 multiplied by one add 𝑟 to the power 𝑡, where 𝑎 is the initial value, 𝑟 is the rate of growth, often given as a percentage but expressed as a decimal, 𝑡 is the number of time intervals, and 𝑦 is the value we get after 𝑡 intervals.
So for our question, the initial value is the population of the city, which is 𝑥. We multiply this by one add the growth rate, which is one add 0.13 because that’s 13 percent expressed as a decimal. And then, we raise it to the 𝑡 power, which is nine for our question because we want the population of the city in nine years’ time. One add 0.13 is just 1.13. And when we work out 1.13 raised to the power nine, we find that it’s 3.004 to three decimal places. We’ll write this with the coefficient first as 𝑦 equals 3.004𝑥.