Here you will compare the population growth of Bentonville, AR to the population decay of Pine Bluff, AR.
In 2000, the population of Bentonville was 20,308 people and Pine Bluff's was 55,068 people.
The population growth rate for Bentonville since 2000 is 5.07%
The population decay rate for Pine Bluff since 2000 is 1.43%
Question 1
Write an exponential function to represent the Bentonville’s population, y, based on the number of years after 2000, x, after a period of exponential growth
The exponential function to represent Bentonville's population growth is:
y = 20308 * (1 + 0.0507)^x
Where:
y = Population of Bentonville after a number of years x
20308 = Population of Bentonville in 2000
0.0507 = Population growth rate
Describe the variables and numbers that you used in your equation.
y = 20308 * (1 + 0.0507)^x
In the equation y = 20308 * (1 + 0.0507)^x:
- y represents the population of Bentonville after a certain number of years x.
- 20308 is the population of Bentonville in the year 2000.
- 0.0507 is the population growth rate, which is expressed as a decimal.
- x is the number of years after the year 2000.
Write an exponential function to represent the Pine Bluff’s population, y, based on the number of years after 2000, x, after a period of population decline. y = 20308 * (1 + 0.0507)^x
The exponential function to represent Pine Bluff's population decline would be:
y = 55068 * (1 - 0.0143)^x
Where:
- y represents the population of Pine Bluff after a certain number of years x.
- 55068 is the population of Pine Bluff in the year 2000.
- 0.0143 is the population decay rate, which is expressed as a decimal.
- x is the number of years after the year 2000.
Explain the similarities and differences between your equations for Bentonville and Pine Bluff.
pine bluff y = 55068 * (1 - 0.0143)^x
Bentonville y = 20308 * (1 + 0.0507)^x
Both equations represent exponential functions that model changes in population over time for two different cities. Here are the similarities and differences between the equations for Bentonville and Pine Bluff:
Similarities:
1. Both equations are exponential functions, where the population (y) changes over time (x).
2. The initial population values for Bentonville (20308) and Pine Bluff (55068) are used in both equations.
3. The variable x represents the number of years after 2000 in both equations.
Differences:
1. The population growth rate is positive in the Bentonville equation (0.0507), reflecting an increase in population over time, while the population decay rate is negative in the Pine Bluff equation (-0.0143), indicating a decrease in population over time.
2. In the Bentonville equation, the growth rate is added to 1, while in the Pine Bluff equation, the decay rate is subtracted from 1. This reflects the direction of change in population for each city.
3. The final population value for Pine Bluff is calculated as a decay from the initial population value, while the final population value for Bentonville is calculated as growth from the initial population value.
During what year will the population of Bentonville first exceed that of Pine Bluff?
During what year will the population of Bentonville be at least twice the size of the population of Pine Bluff?
Show all of your work and explain your steps.
pine bluff y = 55068 * (1 - 0.0143)^x
Bentonville y = 20308 * (1 + 0.0507)^x
To determine the year when the population of Bentonville first exceeds that of Pine Bluff, we need to solve the following inequality:
20308 * (1 + 0.0507)^x > 55068 * (1 - 0.0143)^x
Let's first simplify the equation:
20308 * 1.0507^x > 55068 * 0.9857^x
Now, let's take the natural logarithm of both sides to solve for x:
ln(20308) + x*ln(1.0507) > ln(55068) + x*ln(0.9857)
Now we can solve for x to find the year when the population of Bentonville first exceeds that of Pine Bluff. Calculate x.
Next, we need to find the year when the population of Bentonville is at least twice the size of Pine Bluff. We'll set up the following inequality:
20308 * (1 + 0.0507)^x >= 2 * [55068 * (1 - 0.0143)^x]
Simplify this equation, take the natural logarithm of both sides, solve for x to find the year when Bentonville's population is at least twice the size of Pine Bluff.
Finally, to determine the actual year, you would add the calculated values of x to the base year of 2000.
Doing these steps will provide you with the exact years when the population of Bentonville first exceeds that of Pine Bluff and when Bentonville is at least twice the size of Pine Bluff.