Estimating Population Growth: A 5-Year Analysis

Hey everyone! Let's dive into a cool math problem today. We're going to figure out how a city's population grew over five years. This is a super practical scenario, and understanding how to calculate population growth is a valuable skill. We'll break down the problem step-by-step to make it crystal clear, so stick with me!

Understanding the Population Growth Problem

So, here's the deal: A city's population jumped from 100,000 to 250,000 in just five years. That's a significant increase! The question asks us to find the estimated annual compound percentage increase in the population. Basically, we want to know what percentage the population grew by each year, assuming it grew at the same rate every year. This is a classic compound interest problem, but instead of money, we're dealing with people! Understanding population growth is super important for urban planning, resource allocation, and predicting future needs. Think about it: cities need to plan for schools, hospitals, housing, and infrastructure based on how many people live there. If a city's population is growing rapidly, it needs to be ready to accommodate that growth. Conversely, if a city is losing population, it needs to understand why and adjust its strategies accordingly. This type of analysis helps policymakers make informed decisions. Also, consider the impact on the environment and the strain on existing resources like water, sanitation, and waste management. Rapid population growth can put enormous pressure on these systems, potentially leading to shortages and environmental degradation. Understanding these dynamics helps us to develop sustainable urban development strategies. That is why it is very crucial to understand and calculate the population growth.

Now, let's look at the given options:

• (a) 20%
• (b) 30%
• (c) 10%
• (d) 40%

Our task is to pick the correct one. Don't worry, we'll get there in no time!

Step-by-Step Calculation: Unveiling the Growth Rate

Alright, let's crunch some numbers! To find the annual compound percentage increase, we'll use a formula derived from the compound interest concept. The basic idea is that the population at the end of each year is the population from the previous year, plus the growth. To make it easier to understand, let's break it down in a more accessible way. We can start by considering the ratio of the final population to the initial population. This gives us a sense of the overall growth factor over the five years. Then, we need to take the fifth root of this ratio to find the annual growth factor. Finally, we'll convert the annual growth factor into a percentage to match the answer options.

Here’s the formula we'll use:

Final Population = Initial Population * (1 + r)^n

Where:

• Final Population = 250,000
• Initial Population = 100,000
• r = annual growth rate (what we want to find)
• n = number of years = 5

Let's rearrange the formula to solve for r:

(1 + r)^5 = Final Population / Initial Population

(1 + r)^5 = 250,000 / 100,000

(1 + r)^5 = 2.5

Now, take the fifth root of both sides:

1 + r = (2.5)^(1/5)

1 + r ≈ 1.19

Now, solve for r:

r ≈ 1.19 - 1

r ≈ 0.19

To express this as a percentage, multiply by 100:

r ≈ 0.19 * 100 = 19%

This calculation tells us that the approximate annual growth rate is 19%. This is closest to option (a) 20%. Keep in mind that due to rounding during our calculations, the answers might not be exact. Let's explore the process. First, we determine the total growth factor over the five-year period. This is done by dividing the final population by the initial population. In our case, the population increased by a factor of 2.5 (250,000 / 100,000). Next, since we want to find the annual growth rate, we need to find out what factor the population grew by each year. We do this by taking the fifth root of the total growth factor. The fifth root of 2.5 gives us the annual growth factor, which represents the factor by which the population increased each year. Finally, we convert this annual growth factor into a percentage to get the annual growth rate. This provides a clear and understandable measure of the population's increase. These concepts are fundamental in understanding not only population dynamics but also other areas like financial investments, where compound interest plays a similar role.

Analyzing the Answers: Which Option is the Closest?

So, based on our calculation, the population grew at about 19% annually. Now let’s see which of the provided options is closest. If you've been following along, you've probably already figured it out. Option (a) 20% is the closest to our calculated value of 19%. Options (b), (c), and (d) are all significantly different from our result, so we can confidently eliminate them. Therefore, the correct answer is (a) 20%. This implies that, approximately, the population increased by 20% each year, compounded annually. Let's delve into why the other options are incorrect. Option (b), at 30%, would imply a much more rapid growth rate than observed. If the population grew at 30% per year, the total population would have been far greater than 250,000 after five years. Option (c), at 10%, represents a slower growth rate. If the population only grew by 10% per year, the final population would have been considerably less than 250,000. Option (d), at 40%, indicates an even more extreme growth rate, leading to an unrealistic final population. These examples show how a slight difference in the percentage can lead to a dramatically different outcome over a five-year period, highlighting the sensitivity of compound growth to the rate of increase. Understanding the context of the question and the plausibility of the results is crucial. The choices that are way too high or low should have been immediately discarded.

The Significance of Population Growth Calculations

Population growth calculations are essential because they provide valuable insights that extend beyond just theoretical math problems. They have real-world applications across various fields, including urban planning, economics, environmental science, and public health. Understanding how populations change over time allows for effective resource allocation. For example, knowing the projected population of a city helps local governments plan for the construction of schools, hospitals, and infrastructure like roads and public transportation. This proactive approach ensures that the necessary services and facilities are available to meet the needs of a growing population. These calculations also impact economic planning. Population growth can significantly affect economic growth by influencing labor force size, consumer demand, and market trends. Rapid population growth, if not managed, can place a strain on existing resources such as water and energy. This can lead to increased prices and potential shortages. The impact on the environment should be considered, with a growing population leading to the increased production of waste and greenhouse gasses. This affects ecosystems and contributes to climate change. Additionally, these calculations are critical for public health. They influence resource planning. Understanding how populations change can help public health officials prepare for potential outbreaks of disease and allocate resources effectively. By studying population growth, healthcare providers can allocate enough resources to meet the needs of a growing population. Population studies also help identify specific populations that may need specialized healthcare services.

Conclusion: Mastering the Population Growth Problem

Alright, guys, we did it! We successfully calculated the approximate annual compound percentage increase in the city's population. By understanding the formula and the steps involved, you can now tackle similar problems with confidence. Population growth calculations can be a bit tricky. We started with the basic formula, rearranged it to solve for the growth rate, and then took the fifth root to get the annual rate. The key takeaway is to grasp the concept of compound interest and how it applies to real-world scenarios like population growth. Remember to always double-check your calculations, especially when dealing with percentages. Now you're ready to apply this knowledge to other problems and maybe even impress your friends and family with your math skills! Keep practicing, and you'll become a population growth expert in no time. Thanks for hanging out and working through this problem with me! I hope it was helpful. Keep practicing and exploring other math problems! Until next time, keep learning, and keep growing! This example demonstrates how a simple mathematical concept can have a significant impact on decision-making in various fields. Population growth is just one example, but the underlying principles can be applied to many other scenarios where understanding change over time is essential. Remember to always consider the context of the problem and to think critically about the results.