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):
estimated population
To answer the questions, we need to calculate the annual growth rate and use it to estimate the population in 2020.
1. To calculate the annual growth rate as a percentage, we can use the following formula:
Annual Growth Rate = ((New Population - Old Population) / Old Population) * 100
Given:
Old Population in 2005 = 5,130,632
New Population in 2006 = 6,166,318
Annual Growth Rate = ((6,166,318 - 5,130,632) / 5,130,632) * 100
Plugging in the numbers, we have:
Annual Growth Rate = (1,035,686 / 5,130,632) * 100
Calculating this expression gives us the annual growth rate as a decimal. To convert it to a percentage, simply multiply by 100:
Annual Growth Rate = 0.201790072 * 100 = 20.1790072%
Therefore, the annual growth rate is approximately 20.1790072%.
2. (a) To calculate the estimated population of Arizona in 2020 using discrete exponential growth, we can use the formula:
Estimated Population = Old Population * (1 + Annual Growth Rate)^Number of Years
Given:
Old Population in 2006 = 6,166,318
Annual Growth Rate = 20.1790072%
Number of Years = 14
Estimated Population in 2020 = 6,166,318 * (1 + 0.201790072)^14
Plugging in the numbers, we have:
Estimated Population in 2020 = 6,166,318 * (1.201790072)^14
Calculating this expression, we find that the estimated population in 2020 using discrete exponential growth is approximately 42,210,323.
(b) To calculate the estimated population of Arizona in 2020 using continuous exponential growth, we can use the formula:
Estimated Population = Old Population * e^(Annual Growth Rate * Number of Years)
Given:
Old Population in 2006 = 6,166,318
Annual Growth Rate = 20.1790072%
Number of Years = 14
Estimated Population in 2020 = 6,166,318 * e^(0.201790072 * 14)
Plugging in the numbers, we have:
Estimated Population in 2020 = 6,166,318 * e^(2.8244621)
Calculating this expression, we find that the estimated population in 2020 using continuous exponential growth is approximately 42,939,247.
(c) To calculate the estimated population of Arizona in 2020 using linear growth, we can use the formula:
Estimated Population = Old Population + (Annual Growth Rate * Number of Years)
Given:
Old Population in 2006 = 6,166,318
Annual Growth Rate = 20.1790072%
Number of Years = 14
Estimated Population in 2020 = 6,166,318 + (0.201790072 * 14)
Plugging in the numbers, we have:
Estimated Population in 2020 = 6,166,318 + 2.8244621
Calculating this expression, we find that the estimated population in 2020 using linear growth is approximately 6,168,142.
Finally, to calculate the ratio of growth (new/old) for each scenario, we can divide the estimated population in 2020 by the old population in 2006:
(a) Ratio of Growth (Discrete Exponential Growth) = Estimated Population in 2020 / Old Population in 2006 = 42,210,323 / 6,166,318
(b) Ratio of Growth (Continuous Exponential Growth) = Estimated Population in 2020 / Old Population in 2006 = 42,939,247 / 6,166,318
(c) Ratio of Growth (Linear Growth) = Estimated Population in 2020 / Old Population in 2006 = 6,168,142 / 6,166,318
Calculate these ratios to get the final answers.