Population Growth: Time To Reach 28,000?

by Alex Johnson 41 views

Introduction

In this article, we'll dive into a fascinating problem of population growth. We'll explore how to calculate the time it takes for a town's population to grow from one number to another, given a continuous growth rate. We'll use a mathematical equation to model this growth and then solve it to find the answer. This is a practical application of exponential functions, which are commonly used to model growth and decay in various fields, including biology, finance, and physics. Understanding these concepts is essential for anyone interested in these areas, as well as for anyone who wants to develop their problem-solving skills. This exploration will be both engaging and enlightening, providing you with a clear understanding of how mathematical models can be used to describe and predict real-world phenomena. Population growth is a dynamic process influenced by various factors such as birth rates, death rates, migration, and available resources. These factors interact in complex ways, making it challenging to predict future population trends accurately. However, mathematical models like the exponential growth model provide a simplified framework for understanding and analyzing population dynamics. By making certain assumptions, such as a constant growth rate and unlimited resources, these models can offer valuable insights into the potential trajectory of population growth over time.

Problem Setup

Let's set the stage for our problem. Imagine a town whose population has been steadily increasing. Initially, the town had a population of 20,000 residents. Over time, the population grew to 28,000. We also know that the town's population grows continuously at a rate of 15% per year. Our goal is to figure out how many years it took for the population to grow from 20,000 to 28,000, given this continuous growth rate. The equation that models this situation is given as 20,000e0.15t=28,00020,000 e^{0.15 t} = 28,000, where t represents the number of years the population has been growing. This equation is based on the exponential growth model, which is a mathematical representation of how a quantity increases over time when its growth rate is proportional to its current value. In this case, the population grows at a rate of 15% per year, so the growth rate is 0.15. The constant e is the base of the natural logarithm, which is approximately equal to 2.71828. It is a fundamental mathematical constant that appears in many areas of mathematics and physics. The exponential growth model is a powerful tool for understanding and predicting population growth, but it is important to note that it is based on certain assumptions, such as a constant growth rate and unlimited resources. In reality, these assumptions may not always hold, and the actual population growth may deviate from the model's predictions. Understanding the problem setup is crucial for solving it effectively. By identifying the key information and the goal, we can develop a strategy for tackling the problem. In this case, we know the initial population, the final population, and the growth rate. Our goal is to find the time it takes for the population to grow from the initial value to the final value.

Solving the Equation

Now, let's roll up our sleeves and solve the equation 20,000e0.15t=28,00020,000 e^{0.15 t} = 28,000 to find the value of t. This will tell us how many years it took for the population to grow from 20,000 to 28,000. To solve for t, we'll need to isolate it on one side of the equation. Here's how we'll do it:

  1. Divide both sides by 20,000: This will get rid of the coefficient in front of the exponential term. e0.15t=28,00020,000=1.4e^{0.15 t} = \frac{28,000}{20,000} = 1.4
  2. Take the natural logarithm of both sides: This will undo the exponential function and bring the exponent down. ln⁡(e0.15t)=ln⁡(1.4)\ln(e^{0.15 t}) = \ln(1.4) 0.15t=ln⁡(1.4)0.15t = \ln(1.4)
  3. Divide both sides by 0.15: This will isolate t on one side of the equation. t=ln⁡(1.4)0.15t = \frac{\ln(1.4)}{0.15}

Now, we can use a calculator to find the approximate value of t:

t≈0.336470.15≈2.24t ≈ \frac{0.33647}{0.15} ≈ 2.24 years

Therefore, it took approximately 2.24 years for the town's population to grow from 20,000 to 28,000. Solving exponential equations involves using logarithmic functions to isolate the variable in the exponent. The natural logarithm, denoted as ln, is the inverse function of the exponential function with base e. By taking the natural logarithm of both sides of an exponential equation, we can bring the variable down from the exponent and solve for it. This technique is widely used in various applications, such as modeling population growth, radioactive decay, and compound interest. Understanding how to solve exponential equations is a valuable skill for anyone working with mathematical models of real-world phenomena.

Detailed Steps Explained

Let's break down the steps involved in solving the equation 20,000e0.15t=28,00020,000 e^{0.15 t} = 28,000 even further to ensure clarity. Each step is crucial, and understanding the logic behind each one will help you tackle similar problems in the future. Here's a more detailed explanation:

  1. Divide both sides by 20,000:

    • Why? Our goal is to isolate the exponential term, e0.15te^{0.15t}. Dividing both sides by 20,000 cancels out the 20,000 on the left side, leaving us with just the exponential term.
    • How? 20,000e0.15t20,000=28,00020,000\frac{20,000 e^{0.15 t}}{20,000} = \frac{28,000}{20,000} e0.15t=1.4e^{0.15 t} = 1.4

  2. Take the natural logarithm of both sides:

    • Why? The natural logarithm (ln) is the inverse function of the exponential function with base e. This means that ln⁡(ex)=x\ln(e^x) = x. By taking the natural logarithm of both sides, we can bring the exponent, 0.15t, down from the exponential.
    • How? ln⁡(e0.15t)=ln⁡(1.4)\ln(e^{0.15 t}) = \ln(1.4) 0.15t=ln⁡(1.4)0.15t = \ln(1.4)

  3. Divide both sides by 0.15:

    • Why? We want to isolate t, which represents the number of years. Dividing both sides by 0.15 cancels out the 0.15 on the left side, leaving us with t by itself.
    • How? 0.15t0.15=ln⁡(1.4)0.15\frac{0.15t}{0.15} = \frac{\ln(1.4)}{0.15} t=ln⁡(1.4)0.15t = \frac{\ln(1.4)}{0.15}

  4. Calculate the approximate value of t:

    • Why? We need a numerical answer to understand how many years it took for the population to grow.
    • How? Use a calculator to find the natural logarithm of 1.4 and then divide by 0.15. t≈0.336470.15≈2.24t ≈ \frac{0.33647}{0.15} ≈ 2.24 years

By following these steps carefully, we can solve for t and determine the time it took for the population to grow. Understanding the detailed steps not only helps in solving the current problem but also equips you with the knowledge to tackle similar exponential growth problems in the future. Each step is based on fundamental mathematical principles, and mastering these principles will enhance your problem-solving abilities.

Conclusion

In conclusion, we've successfully calculated that it took approximately 2.24 years for the town's population to grow from 20,000 to 28,000, given a continuous growth rate of 15%. We achieved this by setting up an exponential equation, understanding the properties of exponential and logarithmic functions, and carefully solving for the unknown variable, t. This example demonstrates the power of mathematical models in describing real-world phenomena like population growth. By understanding the underlying principles and techniques, you can apply these concepts to solve a wide range of problems in various fields. Remember, the key is to break down complex problems into smaller, manageable steps and to understand the logic behind each step. With practice and a solid understanding of the fundamentals, you can confidently tackle any mathematical challenge that comes your way. If you're interested in learning more about exponential growth and its applications, check out resources like Khan Academy's Exponential Growth and Decay section.