Use the model below to estimate the average annual growth rate of a certain country's population for 1950, 1988, and 2010, where x is the number of years after 1900.
y = -0.000009x^3 + 0.0025x^2 - 0.201x + 7.979
The estimated average annual growth rate of the country's population for 1950 is ?
To estimate the average annual growth rate for 1950, we need to substitute x = 50 into the equation and calculate the value of y.
y = -0.000009(50)^3 + 0.0025(50)^2 - 0.201(50) + 7.979
Simplifying the equation gives:
y = -0.000009(125000) + 0.0025(2500) - 0.201(50) + 7.979
y = -1.125 + 6.25 - 10.05 + 7.979
y ≈ 3.054
Therefore, the estimated average annual growth rate of the country's population for 1950 is approximately 3.054.
To estimate the average annual growth rate of the country's population for 1950, we need to calculate the derivative of the population function with respect to x and evaluate it at x = 50 (since 1950 is 50 years after 1900).
So, the first step is to find the derivative of the population function:
dy/dx = -0.000027x^2 + 0.005x - 0.201
Next, we substitute x = 50 into the derivative equation:
dy/dx = -0.000027(50)^2 + 0.005(50) - 0.201
= -0.000027(2500) + 0.25 - 0.201
= -0.0675 + 0.25 - 0.201
= -0.0185
Therefore, the estimated average annual growth rate of the country's population for 1950 is approximately -0.0185.
To estimate the average annual growth rate of the country's population for 1950 using the provided model, we need to calculate the population for the years 1949 and 1950 and then find the difference between these two values.
Let's start by substituting x with the number of years after 1900 for the year 1949:
x = 1949 - 1900 = 49
Now, substitute this value into the equation to determine the population for the year 1949:
y = -0.000009(49)^3 + 0.0025(49)^2 - 0.201(49) + 7.979
Calculate the value of y for 1949:
y ≈ -0.000009(117,649) + 0.0025(2,401) - 0.201(49) + 7.979
y ≈ -1.0589 + 6.0025 - 9.8499 + 7.979
y ≈ 3.0737
Now, substitute x with the number of years after 1900 for the year 1950:
x = 1950 - 1900 = 50
Substitute this value into the equation to determine the population for the year 1950:
y = -0.000009(50)^3 + 0.0025(50)^2 - 0.201(50) + 7.979
Calculate the value of y for 1950:
y ≈ -0.000009(125,000) + 0.0025(2,500) - 0.201(50) + 7.979
y ≈ -1.125 + 6.2500 - 10.05 + 7.979
y ≈ 3.054
Now, we can find the difference in population between 1950 and 1949:
Difference = Population in 1950 - Population in 1949
Difference = 3.054 - 3.0737
Difference ≈ -0.0197
Therefore, the estimated average annual growth rate of the country's population for 1950 is approximately -0.0197.