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.

Question

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

Answer 1

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%.