Do some research and find a city that has experienced population growth. Determine its population in January 1st of a certain year. write an exponential function to represents the city's population, y, based on the number of years that pass, x after a period of exponential growth. Describe the variables and numbers that you used in your equation.
One city that has experienced population growth is Austin, Texas. According to the U.S. Census Bureau, the population of Austin on January 1st, 2010 was approximately 790,390.
Let's represent the population of Austin, y, based on the number of years that pass, x after a period of exponential growth. The exponential function can be written as:
y = 790,390 * (1 + r)^x
Where:
- y is the population of Austin after x years
- 790,390 is the initial population of Austin on January 1st, 2010
- r is the annual growth rate of the population (in decimal form)
To find the annual growth rate, we can use the population of Austin on January 1st, 2020, which was approximately 978,908. We can use the formula to find the growth rate:
978,908 = 790,390 * (1 + r)^10
1.2382 = (1 + r)^10
1 + r = 1.2382^(1/10)
1 + r ≈ 1.0221
r ≈ 0.0221
Therefore, the exponential function representing the population of Austin, Texas based on the number of years that pass after a period of exponential growth is:
y = 790,390 * (1.0221)^x