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=-0.0000087x^3+0.00262x^2-0.174+8.172
The estimated average annual growth rate of the country's population for 2010 is
To estimate the average annual growth rate of the country's population for 2010, we need to find the derivative of the given model with respect to x and evaluate it at x = 110 (since 2010 is 110 years after 1900).
Taking the derivative of the model:
dy/dx = -0.0000261x^2 + 0.00524x
Now substituting x = 110 into the derivative:
dy/dx = -0.0000261(110)^2 + 0.00524(110)
= -0.0000261(12100) + 0.5764
= -3.1626 + 0.5764
= -2.5862
The estimated average annual growth rate of the country's population for 2010 is approximately -2.5862.
To estimate the average annual growth rate for the year 2010, we need to find the derivative of the given function. The derivative will give us the rate of change of the population with respect to time. So, let's find the derivative of the function:
dy/dx = -0.0000261x^2 + 0.00524x + 0.00262
Now, to find the growth rate for the year 2010, we substitute x = 110 (as 2010 is 110 years after 1900) into the derivative:
dy/dx (2010) = -0.0000261(110)^2 + 0.00524(110) + 0.00262
= -0.0000261(12100) + 0.00524(110) + 0.00262
= -3.1626 + 0.5744 + 0.00262
= -2.58558
Therefore, the estimated average annual growth rate of the country's population for 2010, based on the given model, is approximately -2.586%.