4. The population of the city of Fredericksburg, Virginia, was approximately 19,360 in 2000 and has been continuously growing at a rate of \( 2.9 \% \) each year. a. Use the formula for population growth to write a function to model this growth. b. Use your function model to predict the population of Fredericksburg in 2020 . c. What value does your function model give for the population of Fredericksburg in the year 1980 ? 5. Use technology to estimate the number of years it would take Fredericksburg to grow to 40,000 people, assuming that the population trend continues.

To model the population growth, we can use the formula \( P(t) = P_0(1 + r)^t \), where \( P_0 \) is the initial population, \( r \) is the growth rate (in decimal), and \( t \) is the number of years since the starting point. Plugging in the values for Fredericksburg, the function becomes \( P(t) = 19360(1 + 0.029)^t \). To predict the population in 2020, we calculate \( t \) as the number of years since 2000, which is 20 years. Substituting in the values, we find \( P(20) = 19360(1.029)^{20} \), resulting in a population of approximately 26,144 in 2020. Now, for the year 1980, we note that this is 20 years before 2000, so \( t = -20 \). Calculating \( P(-20) = 19360(1.029)^{-20} \) yields around 14,607 in 1980. For estimating the time to reach 40,000 people, we set up the equation \( 40000 = 19360(1.029)^t \) and solve for \( t \). Using logarithms or technology can quickly yield an estimate of about 25 years, placing the target year around 2025 based on the continuous growth trend.