Question

The population \(P\) of Alabama (in thousands) for 1995 projected through 2025 can be modeled by \(P=4227(1.0104)^{t},\) where \(t\) is the number of years since \(1995 .\) Find the ratio of the population in 2025 to the population in \(2000 .\) Compare this ratio with the ratio of the population in 2000 to the population in $1995

Short answer

After performing the calculations for the populations in 2000 and 2025 and computing the ratios required, compare the two ratios to answer the problem of whether the population growth rate has remained constant, increased, or decreased from 1995 to 2025.

Step-by-step solution

Calculation of Population in 2000 and 2025

First, you need to compute the population in the year 2000 and in the year 2025. To do this, replace the variable \(t\) in the equation with the corresponding number of years since 1995 for each year of interest. For year 2000, \(t = 2000-1995 = 5\). Similarly, for year 2025, \(t = 2025-1995 = 30\). Then plug-in those values in the model equation \(P=4227(1.0104)^t\).

Calculation of Ratios

To find the ratio of population in 2025 to the population in 2000, divide the population in 2025 by the population in 2000. Repeat the process to get the ratio of the population in 2000 to the population in 1995. Remember, the population in 1995 is simply \(P(t = 0) = 4227\).

Comparing the Ratios

After calculating the two ratios, compare them to determine if the growth of population has remained constant, increased, or decreased.