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 = -0.000009x^3 + 0.0025x^2 - 0.201x + 7.979
The estimated average annual growth rate of the country's population for 2010 is ?
To estimate the average annual growth rate of a certain country's population for 2010 using the given model, we need to find the derivative of the model function with respect to x. The derivative will give us the rate of change of the population over time.
Taking the derivative of the model function y = -0.000009x^3 + 0.0025x^2 - 0.201x + 7.979, we get:
dy/dx = -0.000027x^2 + 0.005x - 0.201
To estimate the average annual growth rate for 2010 (which is 110 years after 1900), we substitute x = 110 into the derivative:
dy/dx = -0.000027(110)^2 + 0.005(110) - 0.201
Simplifying this equation will give us the estimated average annual growth rate for 2010.
To estimate the average annual growth rate of the country's population for 2010, we need to find the derivative of the given function and evaluate it at x = 110 (since 2010 is 110 years after 1900).
The derivative of the function is:
y' = -0.000027x^2 + 0.005x - 0.201
To find the average annual growth rate for 2010, we substitute x = 110 into the derivative:
y' = -0.000027(110)^2 + 0.005(110) - 0.201
= -0.000027 * 12100 + 0.005 * 110 - 0.201
= -3.267 + 0.55 - 0.201
= -2.918
Therefore, the estimated average annual growth rate of the country's population for 2010 is -2.918 percent.
To estimate the average annual growth rate of the country's population for 2010 using the given model, we need to calculate the derivative of the model function.
Step 1: Find the derivative of the function y(x) with respect to x.
dy/dx = d/dx (-0.000009x^3 + 0.0025x^2 - 0.201x + 7.979)
dy/dx = -0.000027x^2 + 0.005x - 0.201
Step 2: Substitute x = 110 into the derivative function to find the growth rate for 2010.
dy/dx = -0.000027(110)^2 + 0.005(110) - 0.201
dy/dx ≈ -0.000027(12100) + 0.005(110) - 0.201
dy/dx ≈ -3.267 + 0.55 - 0.201
dy/dx ≈ -2.918
The estimated average annual growth rate of the country's population for 2010 is approximately -2.918.