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In modeling animal populations, we often encounter situations where the population is not constant but changes over time. This change is typically due to factors such as birth rates, death rates, and migration. For many natural populations, especially under unobstructed conditions, growth can often be described using an exponential model.
This is particularly appropriate when resources are abundant, and the rate of population increase is proportional to the size of the current population. As seen in the exercise, the given animal population grows over time, which we can represent with an exponential model.

• The population starts at a known initial size. In the exercise, the population starts at 500 individuals when time (\(t = 0\)).
• The numbers change according to a specified growth rate, which we will determine.
• The model helps predict future population sizes.

Understanding this model is crucial for predicting long-term population changes and planning conservation strategies that ensure species survival.

To predict future population sizes, identifying the growth rate is essential. This rate captures how quickly or slowly the population size changes over time. Let's delve into how we can calculate this rate using provided data.
In our exercise, after two years (\(t = 2\)), the animal population grows from 500 to 1500. To find the growth rate (\(k\)), we use the exponential growth formula:\[ P(t) = P(0) e^{kt} \] Given:

• \( P(0) = 500 \)
• \( P(2) = 1500 \)

Plugging in these values:\(1500 = 500 e^{2k}\)
First, solve for \( e^{2k} \):\[ e^{2k} = \frac{1500}{500} = 3 \]Taking the natural logarithm of both sides, we have:\[ \ln(3) = 2k \Rightarrow k = \frac{\ln(3)}{2} \]With this growth rate, we can predict how the population will evolve over time. Understanding this process is pivotal for accurately forecasting and managing ecosystems.

Exponential functions are powerful mathematical tools, particularly for modeling and predicting growth in populations. They showcase how quantities grow over time at rates proportional to their current size. In our scenario, we're dealing with an animal population growing exponentially.
Having already determined the growth rate (\(k = \frac{\ln(3)}{2}\)), we can write our population formula:\[ P(t) = 500 e^{\frac{\ln(3)}{2}t} \]Using this formula, we calculated the population size at any future point in time.To find the population at \(t = 5\):

• Plug \(t = 5\) into the equation to get:\[ P(5) = 500 e^{\frac{5 \ln(3)}{2}} \]
• Simplify using the property \(e^{\ln(x)} = x\), hence:\[ P(5) = 500 \, \cdot \, 3^{2.5} \]
• Computing \(3^{2.5}\), we find it to be approximately 15.59, thus:
• \[ P(5) \approx 500 \, \cdot \, 15.59 = 7795 \]

This step illustrates how exponential functions adeptly handle population predictions, catering to biological and ecological studies alike.