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 1988 is ?
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%.
To estimate the average annual growth rate of the country's population for 1988 using the given model, we need to calculate the difference in the population between 1988 and the previous year.
First, we need to convert the year 1988 to the number of years after 1900.
1988 - 1900 = 88
Now, we can substitute this value (88) into the equation:
y = -0.000009(88)^3 + 0.0025(88)^2 - 0.201(88) + 7.979
Calculating this equation will give us the estimated population value for 1988. However, to determine the growth rate, we need to calculate the difference between the population of 1988 and the population of the previous year.
So, we need to calculate the population for the previous year, which is 1987:
x = 1987 - 1900 = 87
Now, substitute this value into the equation:
y = -0.000009(87)^3 + 0.0025(87)^2 - 0.201(87) + 7.979
With both population values obtained, we can calculate the average annual growth rate using the formula:
Growth Rate = (Population_1988 - Population_1987) / Population_1987
Substitute the values in the formula:
Growth Rate = (Population_1988 - Population_1987) / Population_1987
Once you have the values for both populations, calculate the difference and divide it by the population of the previous year to obtain the growth rate.
To estimate the average annual growth rate for 1988 using the given model, we need to find the derivative of the population function with respect to time. The derivative will give us the rate of change of the population, which represents the growth rate.
First, let's find the derivative of the population function:
y = -0.000009x^3 + 0.0025x^2 - 0.201x + 7.979
To find the derivative, we differentiate each term separately:
d/dx (-0.000009x^3) = -0.000027x^2
d/dx (0.0025x^2) = 0.005x
d/dx (-0.201x) = -0.201
d/dx (7.979) = 0
Now, we can sum up the derivatives to find the derivative of the population function:
dy/dx = -0.000027x^2 + 0.005x - 0.201
To estimate the average annual growth rate for 1988, we substitute x = 88 (since 1988 is 88 years after 1900) into the derivative:
dy/dx = -0.000027(88)^2 + 0.005(88) - 0.201
Simplifying the expression, we get:
dy/dx ≈ -0.214
Therefore, the estimated average annual growth rate for the country's population in 1988 is -0.214.