Question

Population of Virginia Table 1.10 gives the population of Virginia for several years. Population of Virginia $$\begin{array}{ll}{\text { Year }} & {\text { Population (thousands) }} \\\ {1998} & {6,901} \\ {1999} & {7,000} \\ {2000} & {7,078} \\ {2001} & {7,193} \\ {2002} & {7,193} \\ {2003} & {7,386}\end{array}$$ (a) Compute the ratios of the population in one year by the population in the previous year. (b) Based on part (a), create an exponential model for the population of Virginia. (c) Use your model in part (b) to predict the population of Virginia in \(2008\).

Short answer

The population ratios are calculated for each year by dividing the population of a given year by the population of the previous year. An exponential growth model is created from these ratios, which can then be used to predict the population in future years, such as 2008 in this exercise.

Step-by-step solution

Compute the Population Ratios

First, calculate the ratio of the population in a given year to the population in the previous year to observe the annual growth. This is done by dividing the population of a given year by the population of the previous year. Repeat this process for each year in the provided table.

Create an Exponential Model

Secondly, derive an exponential model, which predicts population based on year. In general, an exponential model can be written as \( P_0 * \text{growth rate}^{(t-t_{0})} \), where \( P_0 \) is the initial population, growth rate is the yearly growth ratio obtained in the previous step, \( t_{0} \) is the initial year, and \( t \) is the current year.

Use Model to Predict Future Population

Lastly, use the exponential model obtained in step two to predict population for 2008. Substitute \( t = 2008 \) in the model and solve to find the predicted population.