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= -0.0000088x^3+0.00214x^2-0.202x+7.906
The estimated average annual growth rate of the country's population for 1950 is
To find the average annual growth rate for the country's population in 1950, we need to substitute x = 1950 - 1900 = 50 into the model equation and then find the derivative of the equation with respect to x.
y = -0.0000088x^3 + 0.00214x^2 - 0.202x + 7.906
y' = -0.0000088(3x^2) + 0.00214(2x) - 0.202
= -0.0000264x^2 + 0.00428x - 0.202
Now, substitute x = 50 into y'.
y'(50) = -0.0000264(50)^2 + 0.00428(50) - 0.202
= -0.0000264(2500) + 0.214 - 0.202
= -0.066 + 0.214 - 0.202
= -0.054
Therefore, the estimated average annual growth rate of the country's population for 1950 is -0.054 or -5.4%.
To estimate the average annual growth rate of the country's population for 1950, we need to find the derivative of the given equation with respect to x. The derivative of the equation will give us the rate of change of the population.
Taking the derivative of the equation:
dy/dx = -0.0000088 * 3x^2 + 0.00214 * 2x - 0.202
Simplifying further:
dy/dx = -0.0000264x^2 + 0.00428x - 0.202
To find the growth rate for 1950, we need to substitute x = 1950 - 1900 = 50 into the derived equation:
dy/dx = -0.0000264(50)^2 + 0.00428(50) - 0.202
Calculating this value will give us the estimated average annual growth rate for 1950.