Question

The towns of Franklin and Chester post their populations on signs just outside of their towns. The signs are

updated once a year at the beginning of the year.
During year 1, Franklin's sign read "Franklin: Population of 20,000", while Chester's sign read
"Chester: Population of 25,000". Each year, the populations grew. Specifically, Franklin's population
grew by 5% each year, and Chester's population grew by 500 people each year.
What is the first year in which Franklin's sign shows a larger number than Chester's sign?

Answer 1

To find the first year in which Franklin's sign shows a larger number than Chester's sign, we need to compare their populations each year until Franklin's population surpasses Chester's.

Step-by-step explanation:

To find the first year in which Franklin's sign shows a larger number than Chester's sign, we need to compare their populations each year until Franklin's population surpasses Chester's. Let's set up an equation to represent this situation.

Let x be the number of years that have passed since the beginning of the experiment. Franklin's population can be represented as 20,000 * (1 + 0.05)^x, since it grows by 5% each year. Chester's population can be represented as 25,000 + 500 * x, since it grows by 500 people each year.

We need to find the value of x when Franklin's population becomes larger than Chester's population. So, we set up the inequality:

20,000 * (1 + 0.05)^x > 25,000 + 500 * x

To solve this inequality, we can use trial and error or a graphing calculator. We find that x is approximately 48. Therefore, in the 48th year, Franklin's sign will show a larger number than Chester's sign.