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 10 years. The pundit also says that the expected growth will quadruple the state population in 10 years.
(2) Assume that the average annual growth rate can be applied over the next 10 years after 2006.
(a) Calculate the estimated population of Arizona in 2016 using discrete exponential growth.
Now calculate the ratio of growth (new to old):
(b) Calculate the estimated population of Arizona in 2016 using continuous exponential growth.
Now calculate the ratio of growth (new/old):
(c) Calculate the estimated population of Arizona in 2016 using linear growth.
Now calculate the ratio of growth (new/old):
Please understand that no one here will do your work for you. However, we will be happy to read over whatever you come up with and make suggestions and/or corrections.
Please post what you think.
THATS WHAT I DON'T UNDERSTAND THOUGH I DON'T KNOW HOW TO START OF THE PROBLEM?
(1) To calculate the state's annual growth rate as a percentage, we will use the formula:
Annual growth rate = ((final population - initial population) / initial population) * 100
Using the population figures given, we can calculate the growth rate as follows:
Annual growth rate = ((5,130,632 - 6,166,318) / 6,166,318) * 100 = (-1,035,686 / 6,166,318) * 100 ≈ -16.79%
Therefore, the state's annual growth rate is approximately -16.79%.
(2a) To calculate the estimated population of Arizona in 2016 using discrete exponential growth, we will use the formula:
Estimated population = initial population * (1 + growth rate)^years
Given that the expected growth will quadruple the state population in 10 years, the growth rate would be four times the annual growth rate. So, the growth rate for 10 years would be -16.79% * 4 = -67.16%.
Estimated population in 2016 = 6,166,318 * (1 + (-0.6716))^10 ≈ 6,166,318 * (0.3288)^10 ≈ 6,166,318 * 0.0316 ≈ 194,582.40
Therefore, the estimated population of Arizona in 2016 using discrete exponential growth is approximately 194,582.40.
The ratio of growth (new to old) would be 194,582.40 / 6,166,318 ≈ 0.0316.
(2b) To calculate the estimated population of Arizona in 2016 using continuous exponential growth, we will use the formula:
Estimated population = initial population * e^(growth rate * years)
Given the annual growth rate of -16.79%, the growth rate for 10 years would be -16.79% * 10 = -167.9%.
Estimated population in 2016 = 6,166,318 * e^(-0.1679 * 10) ≈ 6,166,318 * e^(-1.679) ≈ 6,166,318 * 0.1878 ≈ 1,157,778.61
Therefore, the estimated population of Arizona in 2016 using continuous exponential growth is approximately 1,157,778.61.
The ratio of growth (new to old) would be 1,157,778.61 / 6,166,318 ≈ 0.1878.
(2c) To calculate the estimated population of Arizona in 2016 using linear growth, we will use the formula:
Estimated population = initial population + (growth rate * years)
Linear growth assumes a constant growth rate. Given the annual growth rate of -16.79%, the growth rate for 10 years would be -16.79% * 10 = -167.9%.
Estimated population in 2016 = 6,166,318 + (-0.1679 * 10) ≈ 6,166,318 - 1,679 ≈ 6,164,639
Therefore, the estimated population of Arizona in 2016 using linear growth is approximately 6,164,639.
The ratio of growth (new to old) would be 6,164,639 / 6,166,318 ≈ 0.9997.
To calculate the state's annual growth rate as a percentage, we can use the formula:
Annual Growth Rate = ((New Population - Old Population) / Old Population) * 100
Given that the old population in 2005 was 5,130,632 and the new population in 2006 was 6,166,318, we can calculate the annual growth rate as follows:
Annual Growth Rate = ((6,166,318 - 5,130,632) / 5,130,632) * 100
Calculating this expression will give us the desired annual growth rate as a percentage.
For the estimated population of Arizona in 2016 using discrete exponential growth, we can use the formula:
Estimated Population = Old Population * (1 + Annual Growth Rate)^Number of Years
Given that we are looking for the estimated population in 2016, which is 10 years after 2006, we can plug in the known values into the formula and calculate the estimated population.
For the ratio of growth (new to old), we can use the formula:
Growth Ratio = New Population / Old Population
For the estimated population of Arizona in 2016 using continuous exponential growth, we can use the formula:
Estimated Population = Old Population * e^(Annual Growth Rate * Number of Years)
Where e is Euler's number (approximately 2.71828).
For the estimated population of Arizona in 2016 using linear growth, we can use the formula:
Estimated Population = Old Population + (Annual Growth Rate * Number of Years)
Let's calculate these values step by step to get the answers.