You are listening to a political pundit discussing the growth of the state's population with a radio talk show host. The host states that the State of Arizona had a population of 6,166,318 in 2006. Our pundit says the 2005 population was 5,130,632.
(1)Calculate the state's annual growth rate as a percentage correct to at least six decimal places. You will need this precision to get the rest of the parts correct. %
The conversation continues about how big the state will be in 14 years. The pundit also says that the expected growth will quadruple the state population in 14 years.
(2) Assume that the average annual growth rate can be applied over the next 14 years after 2006.
(a) Calculate the estimated population of Arizona in 2020 using discrete exponential growth.
Now calculate the ratio of growth (new to old):
(b) Calculate the estimated population of Arizona in 2020 using continuous exponential growth.
Now calculate the ratio of growth (new/old):
(c) Calculate the estimated population of Arizona in 2020 using linear growth.
Now calculate the ratio of growth (new/old):
To calculate the state's annual growth rate as a percentage, we will use the formula:
Annual Growth Rate = ((New Population - Old Population) / Old Population) * 100
Given:
Old Population (2005) = 5,130,632
New Population (2006) = 6,166,318
(1) Calculating the state's annual growth rate:
Annual Growth Rate = ((6,166,318 - 5,130,632) / 5,130,632) * 100
= (1,035,686 / 5,130,632) * 100
≈ 0.201662%
Therefore, the state's annual growth rate is approximately 0.201662%.
Now, let's move on to the estimation of the state's population in 2020.
For (2) assume that the average annual growth rate can be applied over the next 14 years after 2006.
(a) Calculating the estimated population of Arizona in 2020 using discrete exponential growth:
Using the discrete exponential growth formula:
Population (t) = Population (0) * (1 + Growth Rate)^t
Where:
Population (0) = 6,166,318 (Population in 2006)
Growth Rate = 0.201662% (Annual Growth Rate)
t = 14 (Years)
Population (2020) = 6,166,318 * (1 + 0.00201662)^14
≈ 6,940,416
The estimated population of Arizona in 2020 using discrete exponential growth is approximately 6,940,416.
Now, let's calculate the ratio of growth (new to old):
Ratio of growth (new to old) = Population (2020) / Population (2006)
≈ 6,940,416 / 6,166,318
≈ 1.125759
Therefore, the ratio of growth using discrete exponential growth is approximately 1.125759.
(b) Calculating the estimated population of Arizona in 2020 using continuous exponential growth:
Using the continuous exponential growth formula:
Population (t) = Population (0) * e^(Growth Rate * t)
Where:
Population (0) = 6,166,318 (Population in 2006)
Growth Rate = 0.00201662 (Annual Growth Rate)
t = 14 (Years)
Population (2020) = 6,166,318 * e^(0.00201662 * 14)
≈ 6,947,983
The estimated population of Arizona in 2020 using continuous exponential growth is approximately 6,947,983.
Now, let's calculate the ratio of growth (new to old):
Ratio of growth (new to old) = Population (2020) / Population (2006)
≈ 6,947,983 / 6,166,318
≈ 1.126002
Therefore, the ratio of growth using continuous exponential growth is approximately 1.126002.
(c) Calculating the estimated population of Arizona in 2020 using linear growth:
Using the linear growth formula:
Population (t) = Population (0) + (Growth Rate * t)
Where:
Population (0) = 6,166,318 (Population in 2006)
Growth Rate = (6,166,318 * 0.201662%) / 14 (Average annual growth rate)
t = 14 (Years)
Population (2020) ≈ 6,166,318 + [(6,166,318 * 0.00201662) / 14 * 14]
≈ 6,166,318 + (12,427.7719256)
≈ 6,178,746
The estimated population of Arizona in 2020 using linear growth is approximately 6,178,746.
Now, let's calculate the ratio of growth (new to old):
Ratio of growth (new to old) = Population (2020) / Population (2006)
≈ 6,178,746 / 6,166,318
≈ 1.002016
Therefore, the ratio of growth using linear growth is approximately 1.002016.
To calculate the annual growth rate, we can use the formula:
Annual growth rate = (Current population - Previous population) / Previous population * 100
(1) Given that the population in 2006 was 6,166,318 and in 2005 was 5,130,632, we can plug these values into the formula:
Annual growth rate = (6,166,318 - 5,130,632) / 5,130,632 * 100
Calculating this value gives us the annual growth rate as a percentage.
Now, let's move on to the second part of the question.
(2a) To calculate the estimated population of Arizona in 2020 using discrete exponential growth, we can use the formula:
Estimated population = Initial population * (1 + growth rate)^years
Given that the initial population in 2006 was 6,166,318 and the growth rate was calculated in the previous step, we can now calculate the estimated population in 2020 (after 14 years):
Estimated population in 2020 (discrete exponential growth) = 6,166,318 * (1 + annual growth rate/100)^14
(2b) To calculate the estimated population of Arizona in 2020 using continuous exponential growth, we can use the formula:
Estimated population = Initial population * e^(growth rate * years)
Here, "e" represents the mathematical constant approximately equal to 2.71828. Using the same values as before, we can calculate the estimated population in 2020 (after 14 years):
Estimated population in 2020 (continuous exponential growth) = 6,166,318 * e^(annual growth rate/100 * 14)
(2c) To calculate the estimated population of Arizona in 2020 using linear growth, we assume a constant growth rate. We can use the formula:
Estimated population = Initial population + growth rate * years
Given that the initial population in 2006 was 6,166,318 and the growth rate remains the same as the annual growth rate previously calculated, we can calculate the estimated population in 2020 (after 14 years):
Estimated population in 2020 (linear growth) = 6,166,318 + annual growth rate/100 * 14
After calculating each of these estimated population values in 2020, we can then calculate the ratio of growth by taking the new estimate and dividing it by the old population estimate from 2006.