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When examining exponential growth, the concept of population size is crucial. Population size, denoted as \( N_{t} \), is the total number of individuals present in a given population at a specific time \( t \).
As time progresses, the population's size can increase, decrease, or remain constant, depending entirely on the growth factors acting upon it. For this exercise, we're looking at an exponential growth scenario, where each unit of time results in a doubling of the population.
It's vital to start with an initial population, known as \( N_{0} \). In this case, the initial population size is 40 individuals at time zero. This base number is foundational because it serves as the starting point from which growth is calculated. The formula \( N_{t} = 40 \times 2^{t} \) helps us understand how the population size evolves over time, given the consistent growth pattern of doubling.

Understanding the growth rate is central to mastering exponential growth models. The growth rate, denoted as \( r \), determines how quickly or slowly a population expands over a given timeframe. For populations that double every unit of time, such as in this problem, the growth rate is derived from the doubling condition.
The primary equation \( N_{1} = N_{0} \times e^{r \times 1} = 2 \times N_{0} \) showcases that after one time unit, the population should be double its initial size. By setting \( 80 = 40 \times e^{r} \), we simplified and solved to find that \( e^{r} = 2 \). Consequently, this leads us to \( r = \ln(2) \), demonstrating that the population grows at a rate equivalent to the natural logarithm of 2.
This approach highlights the direct relationship between growth conditions and how they mathematically translate into a specific growth rate. Understanding and calculating this rate is essential for predicting future population sizes accurately.

Doubling time is the period it takes for a population to double in size. It’s a straightforward but essential metric when evaluating exponential growth dynamics. The idea that a population continually doubles aligns perfectly with the formula and the conditions set in this exercise.
For a population where the size doubles every unit of time, the doubling time is precisely one unit. This is reflected in our simplified population equation \( N_{t} = 40 \times 2^{t} \), which clearly indicates that each progression in \( t \) (time) signifies a doubling.
The exponential nature of this growth, where \( 40 \times 2^{t} \) doubles per time period, confirms that the doubling time is aligned with the \( r = \ln(2) \) growth rate. Calculating such growth offers insights into understanding rapidly increasing populations and the exponential timescales on which they operate. Recognizing the doubling time can impact strategies in fields like biology, ecology, and economics, where managing growth effectively is crucial.