Population Growth: When Does It Reach 11,572?

Let's dive into a fascinating problem about population growth! We'll explore how to use a simple equation to predict when a city's population will reach a specific number. This is a common type of problem in mathematics, particularly in algebra, and it has real-world applications in urban planning, resource management, and demographics. So, grab your thinking caps, and let's get started!

Understanding the Equation: Population Prediction

At the heart of our problem is the equation $P = 112t + 10900$. This equation is a mathematical model that describes how the population, represented by P, of a small city changes over time. The variable t represents the number of years after the year 1956. Let's break down each part of the equation to understand what it tells us:

• P: This represents the total population of the city at a specific time.
• 112: This is the coefficient of t and represents the annual population increase. In other words, the population grows by 112 people each year.
• t: This is the variable that represents the number of years after the base year, 1956. For example, if t is 10, we are looking at the year 1966 (1956 + 10).
• 10900: This is the constant term and represents the initial population of the city in the year 1956 (when t = 0).

This equation is a linear equation, which means that the population grows at a constant rate. Linear equations are widely used to model various real-world phenomena, from simple interest calculations to the motion of objects in physics. Understanding the components of this equation is crucial for predicting the city's future population. Think of it as a roadmap that shows us where the population is heading year after year. By analyzing the equation, we can gain valuable insights into the city's growth trajectory and plan accordingly.

The Question: Reaching a Specific Population

Now that we understand the equation, let's focus on the question we need to answer: When will the population reach 11,572 people? This is a specific target population that we want to predict. To find the answer, we need to use the equation $P = 112t + 10900$ and substitute the target population (11,572) for P. This will give us an equation that we can solve for t, which represents the number of years after 1956 when the population reaches the target. This is a classic application of algebra, where we use equations to model real-world situations and solve for unknown variables. The ability to manipulate equations and solve for unknowns is a fundamental skill in mathematics and has countless practical applications. This scenario perfectly illustrates how math can be used to make predictions and inform decision-making in various fields.

Solving for t: Finding the Year

Here's how we solve for t:

1. Substitute the target population (11,572) for P in the equation: $11572 = 112t + 10900$
2. Subtract 10900 from both sides of the equation to isolate the term with t: $11572 - 10900 = 112t$ $672 = 112t$
3. Divide both sides of the equation by 112 to solve for t: $t = 672 / 112$ $t = 6$

So, we have found that t = 6. This means that the population will reach 11,572 people 6 years after 1956. But the question asks for the year, not the number of years after 1956. Therefore, we need to add 6 to 1956 to find the year when the population reaches 11,572.

Determining the Year: The Final Step

To find the year when the population reaches 11,572, we simply add the value of t (which is 6) to the base year (1956):

$Year = 1956 + 6 = 1962$

Therefore, the population of the city is predicted to reach 11,572 people in the year 1962. This is the final answer to our problem. By using the given equation and applying basic algebraic principles, we have successfully predicted a future population milestone for the city. This exercise highlights the power of mathematical modeling in forecasting trends and providing valuable insights for planning and decision-making.

Conclusion: The Power of Prediction

In conclusion, by using the equation $P = 112t + 10900$, we determined that the population of the city would reach 11,572 people in the year 1962. This problem demonstrates the power of mathematical models in predicting real-world phenomena. Linear equations, like the one we used, are fundamental tools in various fields, including demographics, economics, and engineering. Understanding how to use these equations to make predictions is a valuable skill.

This exercise also highlights the importance of breaking down a problem into smaller, manageable steps. We started by understanding the equation and its components, then identified the target population, solved for the unknown variable (t), and finally, calculated the year when the population would reach the target. This step-by-step approach is a key strategy for tackling complex problems in mathematics and other disciplines.

Remember, mathematics is not just about numbers and equations; it's a powerful tool for understanding and predicting the world around us. By mastering these skills, you can gain valuable insights into various aspects of life, from population growth to financial planning.

For further exploration of population growth models and their applications, you can visit reputable resources such as the U.S. Census Bureau. This will provide you with a deeper understanding of how mathematical models are used in real-world scenarios and how they can help us make informed decisions about the future.