U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:


a. Assume t = 0 corresponds to 2005 and that the population growth is exponential for the first ten years; that is, between 2005 and 2015, the population is given by P(t) = P(0)exp(rt). Estimate the growth rate r using this assumption.

Verified step by step guidance

1. Identify the given information: the population at time \(t=0\) (year 2005) is \(P(0) = 296\) million, and at \(t=10\) (year 2015) it is \(P(10) = 321\) million.
2. Recall the exponential growth model formula: \(P(t) = P(0) \times \exp(rt)\), where \(r\) is the growth rate we want to find.
3. Substitute the known values into the formula for \(t=10\): \$321 = 296 \times \exp(10r)$.
4. Isolate the exponential term by dividing both sides by 296: \(\frac{321}{296} = \exp(10r)\).
5. Take the natural logarithm of both sides to solve for \(r\): \(\ln\left(\frac{321}{296}\right) = 10r\), then express \(r\) as \(r = \frac{1}{10} \ln\left(\frac{321}{296}\right)\).

Key Concepts

Exponential Growth Model

Exponential growth describes a process where the rate of change of a quantity is proportional to its current value, leading to rapid increase over time. It is modeled by the equation P(t) = P(0)exp(rt), where P(0) is the initial population, r is the growth rate, and t is time. This model applies well to populations growing without constraints over short periods.

Growth Rate Estimation

The growth rate r in an exponential model quantifies how quickly the population increases. It can be estimated by rearranging the exponential formula using known population values at two different times: r = (1/t) * ln(P(t)/P(0)). This calculation requires understanding logarithms and interpreting the time interval correctly.

Logarithmic Functions and Natural Logarithm

Logarithms, especially the natural logarithm (ln), are the inverse of exponential functions and are essential for solving equations involving exponentials. In growth rate estimation, taking the natural log linearizes the exponential equation, allowing for straightforward calculation of r from population data.