Population Growth: Estimating The Limit Of P(t)

Hey guys! Let's dive into a super interesting topic today: population growth. We're going to break down a specific function that models how a city's population changes over time. This function is $P(t) = 0.9(1.5)^t$, where t represents time in years. Our main goal is to estimate the limit of this function as t approaches infinity. In simpler terms, we want to figure out what happens to the population in the long run. Will it explode, stabilize, or maybe even decline? Let's find out!

Deciphering the Population Growth Function

Before we jump into the limit, let's make sure we fully understand what this function is telling us. The function $P(t) = 0.9(1.5)^t$ is an exponential growth model. Exponential growth is characterized by a quantity increasing at a rate proportional to its current value. Think of it like a snowball rolling down a hill – it gets bigger and faster as it goes.

• Initial Population: The number 0.9 in the equation represents the initial population at time t = 0. This is like our starting point. Imagine this city starts with a population of 0.9 million people. Remember, in population models, the units are often in millions, so 0.9 represents 900,000 people.

• Growth Factor: The number 1.5 is the growth factor. This tells us how much the population multiplies each year. A growth factor of 1.5 means the population increases by 50% every year. This is a pretty significant growth rate!

• Time Variable: The variable t represents time in years. So, if we plug in t = 1, we're looking at the population after one year; t = 10, after ten years, and so on.

• Exponential Term: The term $(1.5)^t$ is the heart of the exponential growth. As t gets larger, this term grows very, very quickly. This is what drives the rapid increase in population over time.

Understanding these components is crucial. Exponential growth is a powerful concept, and it's used to model all sorts of things, from bacterial growth to financial investments. In our case, it helps us understand how a city's population can change dramatically over time. It's important to recognize that this model is a simplification of reality. Real-world population growth is affected by many factors, such as resource availability, migration patterns, and birth and death rates. However, this function gives us a valuable tool for making estimates and understanding the general trend.

Think of it like this: if the growth factor were less than 1 (say, 0.9), the population would decrease over time. If the growth factor were exactly 1, the population would stay constant. But because our growth factor is 1.5, which is greater than 1, we know the population is going to increase. The question is, how quickly and to what extent?

Evaluating the Limit as Time Approaches Infinity

Now, let's get to the core of the question: estimating $\lim_{t \to \infty} P(t)$. This notation might look intimidating, but it's actually quite straightforward. It's asking us: "What happens to the population, P(t), as time, t, goes on forever?" or, more practically, "What is the population trend in the very long run?"

To figure this out, we need to think about the behavior of our function $P(t) = 0.9(1.5)^t$ as t gets incredibly large. The key here is the exponential term, $(1.5)^t$. As we discussed earlier, exponential functions grow very rapidly when the base (in this case, 1.5) is greater than 1.

Let's consider what happens when we plug in some large values for t:

• If t = 10, P(10) = 0.9 * (1.5)^10 ≈ 51.53
• If t = 20, P(20) = 0.9 * (1.5)^20 ≈ 2940.74
• If t = 30, P(30) = 0.9 * (1.5)^30 ≈ 167,772.16

See the trend? The population is growing at an accelerating rate. It's not just increasing; it's increasing faster and faster as time goes on. This is the hallmark of exponential growth.

As t approaches infinity, $(1.5)^t$ also approaches infinity. When we multiply infinity by a positive number (like 0.9), we still get infinity. Therefore, the limit of P(t) as t approaches infinity is infinity. $\lim_{t \to \infty} P(t) = \infty$ This means that, according to this model, the population of the city will grow without bound in the long run. Of course, in reality, there are limits to population growth. Resources like food, water, and space are finite. However, mathematically, this model tells us that if the growth rate remains constant at 50% per year, the population will increase dramatically over time.

It's crucial to understand the implications of this result. While models like this are useful for making predictions, they are based on certain assumptions. In this case, the model assumes that the growth rate of 50% per year will continue indefinitely. This is unlikely to be the case in the real world. Factors like resource scarcity or changes in birth and death rates could affect the growth rate over time.

Why the Other Options Are Incorrect

Let's quickly look at why the other answer choices are incorrect:

• A. 1.35: This is the result of multiplying 0.9 by 1.5 (0.9 * 1.5 = 1.35). While this might seem like a reasonable number, it doesn't account for the exponential growth over time. It's essentially just looking at the population after one year, not the long-term trend.

• B. 0: This would imply that the population eventually dies out. However, our growth factor is greater than 1, so the population is increasing, not decreasing.

• C. 0.9: This is the initial population at time t = 0. While it's a valid data point, it doesn't tell us anything about the long-term trend of the population. It's like knowing where you started a journey but not knowing where you're going.

Real-World Implications and Limitations

Now, let's take a step back and think about the real-world implications of our findings. While a population growing without bound might seem like a distant theoretical scenario, it highlights some important considerations for urban planning and resource management.

If a city's population is growing rapidly, planners need to think about things like housing, infrastructure (roads, public transportation), and access to resources like water and energy. Failure to plan for growth can lead to overcrowding, strain on resources, and a decline in the quality of life for residents.

However, it's also crucial to remember the limitations of our model. As mentioned earlier, exponential growth cannot continue indefinitely in the real world. Eventually, limiting factors will come into play. These factors could include:

• Resource Scarcity: Limited access to food, water, and energy can constrain population growth.

• Environmental Constraints: Pollution, climate change, and other environmental factors can impact population health and survival.

• Social and Economic Factors: Changes in birth rates, death rates, migration patterns, and economic conditions can all affect population growth.

Therefore, while our model provides a valuable estimate of potential population growth, it's essential to consider these other factors when making real-world decisions. Population models are just one tool in the toolbox; they need to be used in conjunction with other data and expert judgment.

In conclusion, guys, we've seen that the function $P(t) = 0.9(1.5)^t$ predicts that the population of the city will grow without bound as time approaches infinity. This underscores the power of exponential growth, but also reminds us of the importance of considering real-world limitations and planning for sustainable growth.

So, the correct answer is D. ∞. Keep exploring and stay curious!

Key Takeaways

• The function $P(t) = 0.9(1.5)^t$ represents exponential growth.
• The growth factor of 1.5 indicates a 50% annual increase in population.

• \lim_{t \to \infty} P(t) = \infty$, meaning the population grows without bound in this model.

• Real-world population growth is subject to limiting factors not captured in this model.
• Understanding population models is crucial for urban planning and resource management.