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 = - 0.0000086x^3 + 0.00223x^2 - 0.171x + 8.086

The estimated average annual growth rate of the country's population for 1950 is ?

Answer 1

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.