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 = - 0.0000086x^3 + 0.00223x^2 - 0.171x + 8.086
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 find the derivative of the population function with respect to time (x), which will give us the rate of change of the population at any given year.
Taking the derivative of the population function:
dy/dx = - 3(0.0000086)x^2 + 2(0.00223)x - 0.171
Now we need to evaluate the derivative at the year 1950, which is 1950 - 1900 = 50.
dy/dx = - 3(0.0000086)(50)^2 + 2(0.00223)(50) - 0.171
Simplifying the equation:
dy/dx = - 3(0.0000086)(2500) + 2(0.00223)(50) - 0.171
dy/dx = -0.645 + 0.223 - 0.171
dy/dx = -0.593
Therefore, the estimated average annual growth rate of the country's population for 1950 is approximately -0.593.
To estimate the average annual growth rate of the country's population for 1950, we need to evaluate the rate of change of the population function at that particular year.
Given that x represents the number of years after 1900, we can calculate the value of x for 1950 as follows:
x = 1950 - 1900 = 50
Substituting this value of x into the population function:
y = - 0.0000086(50)^3 + 0.00223(50)^2 - 0.171(50) + 8.086
Simplifying the equation:
y ≈ - 0.0000086(125000) + 0.00223(2500) - 0.171(50) + 8.086
y ≈ - 1.075 + 5.575 - 8.55 + 8.086
y ≈ 4.036
Therefore, the estimated average annual growth rate of the country's population for 1950 is approximately 4.036.
To estimate the average annual growth rate of the country's population for 1950, we need to calculate the derivative of the given function with respect to x. The derivative will give us the rate of change of the function at any point, which in this case represents the average annual growth rate.
Taking the derivative of the given function, we get:
dy/dx = -0.0000086 * 3x^2 + 0.00223 * 2x - 0.171
Simplifying this expression gives us:
dy/dx = -0.0000258x^2 + 0.00446x - 0.171
Now, to find the average annual growth rate for 1950, we substitute x = 1950 - 1900 = 50 into the derivative expression:
dy/dx = -0.0000258(50)^2 + 0.00446(50) - 0.171
Calculating this expression will give us the estimated average annual growth rate for 1950.