Estimating Population Growth In Centerville To 2020

by Scholario Team 52 views

Hey guys! Ever wondered how we can predict population growth? It's a fascinating topic, and today, we're diving into a real-world example using a cool mathematical model. We're going to figure out what the population of Centerville might be in 2020, given its population in 1910 and 1920. Buckle up, because we're about to use some math magic!

Understanding Exponential Growth

Before we jump into the numbers, let's quickly chat about exponential growth. This type of growth happens when a quantity increases at a rate proportional to its current value. Think of it like a snowball rolling down a hill – it gets bigger and bigger as it goes! In population terms, exponential growth means the more people there are, the faster the population increases. This is because more people can have more children, leading to even more people, and so on.

The formula we'll be using to model this is: $f(t) = Pe^{rt}$. Don't worry, it's not as scary as it looks! Let's break it down:

  • f(t): This represents the population at a specific time t.
  • P: This is the initial population, or the population at the very beginning of our observation.
  • e: This is a special number in math called Euler's number, approximately equal to 2.71828. It's the base of the natural logarithm and pops up in all sorts of interesting places!
  • r: This is the growth rate, which tells us how quickly the population is increasing. It's usually expressed as a decimal (e.g., 0.05 for 5% growth).
  • t: This is the time that has passed since the initial observation.

So, basically, this formula says that the population at any given time is equal to the initial population multiplied by e raised to the power of the growth rate times the time. Cool, right?

This model assumes consistent growth conditions, which may not always be entirely true in real-world scenarios due to various factors like resource availability, migration, and unforeseen events. However, it provides a solid estimate when we assume that growth is primarily driven by the population size itself.

Setting Up the Problem: Centerville's Population

Okay, let's get back to Centerville! We know the population in 1910 was 4,200, and in 1920, it was 5,000. Our goal is to estimate the population in 2020. To do this, we need to figure out the exponential growth rate (r) that fits this data. Here’s how we'll tackle it:

  1. Identify the knowns:
    • Initial population (P): 4,200 (population in 1910)
    • Population at time t (f(t)): 5,000 (population in 1920)
    • Time elapsed (t): 10 years (1920 - 1910)
  2. Plug the values into our formula:

    • 5000=4200∗e10r5000 = 4200 * e^{10r}

  3. Solve for r (the growth rate): This will involve a bit of algebra, but we can do it!

Solving for the Growth Rate (r)

Alright, let's find that growth rate! We have the equation: $5000 = 4200 * e^{10r}$.

First, we need to isolate the exponential term. We can do this by dividing both sides of the equation by 4200:

50004200=e10r\frac{5000}{4200} = e^{10r}

This simplifies to:

1.190476≈e10r1.190476 ≈ e^{10r}

Now, to get rid of that pesky e, we'll use the natural logarithm (ln). Remember, the natural logarithm is the inverse of the exponential function with base e. So, if we take the natural log of both sides, we get:

ln(1.190476)≈ln(e10r)ln(1.190476) ≈ ln(e^{10r})

Using the property of logarithms that $ln(e^x) = x$, we can simplify the right side:

ln(1.190476)≈10rln(1.190476) ≈ 10r

Calculating the natural log of 1.190476 gives us approximately 0.1744.

0.1744≈10r0.1744 ≈ 10r

Finally, to solve for r, we divide both sides by 10:

r≈0.174410r ≈ \frac{0.1744}{10}

r≈0.01744r ≈ 0.01744

So, the annual growth rate, r, is approximately 0.01744, or 1.744%. That means Centerville's population was growing at a rate of about 1.744% per year during that decade.

Predicting the Population in 2020

Awesome! We've found the growth rate. Now, we can use it to predict Centerville's population in 2020. Remember our formula: $f(t) = Pe^{rt}$.

This time, our knowns are:

  • Initial population (P): 4,200 (population in 1910)
  • Growth rate (r): 0.01744 (calculated above)
  • Time elapsed (t): 110 years (2020 - 1910)

Let's plug these values into the formula:

f(110)=4200∗e(0.01744∗110)f(110) = 4200 * e^{(0.01744 * 110)}

First, we calculate the exponent:

0.01744∗110≈1.91840. 01744 * 110 ≈ 1.9184

Now, we raise e to that power:

e1.9184≈6.811e^{1.9184} ≈ 6.811

Finally, we multiply by the initial population:

f(110)≈4200∗6.811f(110) ≈ 4200 * 6.811

f(110)≈28606.2f(110) ≈ 28606.2

Since we need to round to the nearest person, we get approximately 28,606 people.

Centerville's Estimated Population in 2020

Drumroll, please! Based on our calculations using the exponential growth model, we estimate that the population of Centerville in 2020 would be approximately 28,606 people. Wow, that's quite a jump from 4,200 in 1910!

It's important to remember that this is just an estimate. Real-world population growth can be affected by many factors, and this model assumes a constant growth rate, which might not be perfectly accurate over such a long period. However, it gives us a pretty good idea of how the population might have changed if it continued to grow at a similar rate.

Why This Matters: The Power of Mathematical Modeling

Isn't it cool how we can use math to predict things like population growth? This is the power of mathematical modeling! By understanding the principles behind exponential growth, we can make informed estimates about future trends. This kind of analysis is used in all sorts of fields, from urban planning to economics to environmental science. So, next time you hear about population projections or growth forecasts, you'll know a little bit about the math that goes into them!

Keep exploring, keep learning, and remember, math can be super fun and useful. You guys rock!