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 1950 is_____
To estimate the average annual growth rate of the country's population for 1950, we need to find the value of y when x is equal to 50 (since 1950 is 50 years after 1900).
Substituting x = 50 into the equation:
y = -0.0000087(50)^3 + 0.00262(50)^2 - 0.174(50) + 8.172
Calculating the equation:
y = -0.0000087(125,000) + 0.00262(2,500) - 0.174(50) + 8.172
y ≈ -1.0875 + 6.55 - 8.7 + 8.172
y ≈ 4.855
The estimated average annual growth rate of the country's population for 1950 is approximately 4.855.
To estimate the average annual growth rate of the country's population for 1950, we need to find the derivative of the given model with respect to x. The derivative will give us the rate of change of the population at any given point.
Taking the derivative of the model:
dy/dx = -0.0000087*3x^(3-1) + 0.00262*2x^(2-1)
Simplifying:
dy/dx = -0.0000261x^2 + 0.00524x
Now, substitute x = 1950 - 1900 = 50 into the derivative to get the estimated average annual growth rate for 1950:
dy/dx = -0.0000261(50)^2 + 0.00524(50)
= -0.0000261(2500) + 0.262
= -0.06525 + 0.262
= 0.19675
Therefore, the estimated average annual growth rate of the country's population for 1950 is approximately 0.19675.