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 = -0.000009x^3 + 0.0025x^2 - 0.201x + 7.979

The estimated average annual growth rate of the country's population for 1988 is ?

Answer 1

To find the average annual growth rate for 1988, we need to calculate the slope of the population growth function at that point. In other words, we need to find the derivative of the function with respect to x and evaluate it at x = 1988 - 1900 = 88.

To find the derivative, we can differentiate each term of the function separately:

dy/dx = d/dx (-0.000009x^3) + d/dx (0.0025x^2) - d/dx(0.201x) + d/dx(7.979)

Simplifying each term, we have:

dy/dx = -0.000027x^2 + 0.005x - 0.201

Now, we can evaluate the derivative at x = 88:

dy/dx at x = 88 = -0.000027(88)^2 + 0.005(88) - 0.201

Calculating this expression:

dy/dx at x = 88 = -0.213936 + 0.44 - 0.201

dy/dx at x = 88 = 0.025064

Therefore, the estimated average annual growth rate of the country's population for 1988 is approximately 0.025064, or 2.5064%.