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When dealing with exponential growth, a recursive formula provides a way to find the next term based on its predecessor. In this scenario, we are looking at a bacteria population that starts at 300 and grows exponentially. The recursive formula is great when you want to calculate step-by-step and see how each new population figure depends on the previous one.
Here's how the recursive formula works in our example: each hour, the population grows by multiplying by a factor of the growth rate, expressed as the continuous compounding factor. We already calculated that the growth rate is approximately 0.142. So, we get the recursive formula:

• \( P_{n+1} = P_n \times e^{0.142} \)
• Here, \( P_{n+1} \) is the population after the next hour, and \( P_n \) is the current population.

This formula lets us compute the population incrementally, making it easy to track what happens at each period.

Unlike a recursive formula, an explicit formula allows you to jump directly to the calculation of the population size at any given point in time without the need to calculate every intermediary step. This form of the equation is particularly helpful for predicting future events and populations over longer time frames.

In the context of our bacteria, the explicit formula uses the initial population and growth rate to determine the population size at any given hour, \( t \). The explicit formula derived from the problem is given by:

• \( P(t) = 300 \times e^{0.142t} \)
• Where \( P(t) \) is the population at time \( t \), \( 300 \) is the initial number of bacteria.
• The term \( e^{0.142t} \) captures the effect of continuous growth over time \( t \).

This formula is flexible and powerful for quickly calculating the population at any point, such as 24 hours for a daily estimate.

The growth rate in an exponential growth problem is a crucial component as it determines how quickly the population increases. It's often presented as a constant that modifies the exponential function, enabling predictions at any time in the future.

In our bacteria problem, the growth rate \( r \) was found by utilizing the exponential growth model. This required us to solve the equation using the initial and observed population at a given time:

• We identified that in 4 hours, the population grew from 300 to 500.
• Using the formula \( \frac{500}{300} = e^{4r} \), and then solving \( r = \frac{\ln(\frac{5}{3})}{4} \), we found \( r \approx 0.142 \).
• This rate tells you how quickly the bacteria multiply, and directly feeds into both the recursive and explicit formulas.

Understanding this growth rate allows for accurate predictions of future population sizes and is key to understanding the behavior of exponential growth.

Population growth is a fascinating and complex topic, especially when modeled exponentially, as in our bacteria culture example. When populations grow exponentially, they increase by a set percentage over equal time intervals. This differs dramatically from linear growth, which adds a constant number each interval.

For the bacteria, we started with 300, and after 24 hours, the population soared to approximately 9054, showcasing exponential growth's rapid escalation. Key takeaway points include:

• Exponential growth causes populations to increase much faster over time compared to linear growth.
• The concept helps answer questions like how long it will take for a population to reach a particular size, such as tripling, which we calculated to be about 7.63 hours for our bacteria.
• In applications, understanding exponential growth helps predict trends, guide resource allocation, and manage environments sustainably.

Grasping the fundamentals of population growth empowers us to make informed decisions based on the modeled predictions.