Problem 68 - Algebra 1: Concepts and Skills

Question

Use the following information. Population estimates for the 1800 s lead a student to model the population of the United States by \(P=5,500,400+683,300 t^{2},\) where \(t=0,1,2,3, \ldots\) represents the years \(1800,1810,1820,1830, \ldots\). Use this model to estimate the year in which the United States population reached 50 million.

Step-by-Step Solution

Answer

The United States population reached 50 million around the year 1811 according to the model.

Step 1: Understand the Model

The model used to estimate the population of the United States is given by \(P=5,500,400+683,300 t^{2}\), where \(t\) represents time in increments of 10 years starting from 1800. This means that for each unit increase of \(t\), we move 10 years forward from 1800.

Step 2: Set up the Equation

It is known that the population is 50 million (or 50,000,000). This value needs to be plugged into the model equation for \(P\) to solve for \(t\). So the equation will be \(50,000,000=5,500,400+683,300 t^{2}\).

Step 3: Solve for t

First, isolate \(t^{2}\) by moving 5,500,400 to the other side to get \(44,499,600 = 683,300 t^{2}\). Divide both sides by 683,300 to find \(t^{2}\). After calculation, \(t^{2}\) is approximately 65.12.

Step 4: Calculate t

Finally, to calculate \(t\), take the square root of 65.12. The value of \(t\) is approximately 8.07. Remember that \(t\) is in 10-year increments starting from 1800.

Step 5: Find the Year

To find the corresponding year, multiply \(t\) by 10 and add the result to 1800. So the approximate year when the US population reached 50 million is 1800 + 8.07*10, which is approximately 1811.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that generally have the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). They are called 'quadratic' because \(quadra\) means 'square' in Latin; the highest power of the variable \(x\) is a square. In the population model for the United States, the equation \(P = 5,500,400 + 683,300t^{2}\) is an example of a quadratic equation where the variable \(t\) is squared.

To solve a quadratic equation, you can factor it, complete the square, use the quadratic formula, or graph the function and look for where it intersects the \(x\)-axis. In the context of our model, solving for \(t\) when \(P = 50,000,000\) involves simplifying and rearranging the equation to isolate and find the value of \(t^{2}\). Understanding quadratic equations is crucial for this exercise because it allows us to predict and analyze changes over time, such as the growth of a population.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world situations to predict future events, understand trends, or analyze systems. Models are simplifications of reality that can come in many forms, such as equations, functions, or simulations. In this case, the population estimate of the United States was modeled using a quadratic equation.

The model \(P = 5,500,400 + 683,300t^{2}\) allows us to estimate the US population in a given year by plugging the time \(t\) into the equation. These kinds of models are powerful tools in many fields, from environmental science to economics, as they can provide insight into complex processes. However, it's important to remember that models are based on the data available and assumptions at the time; they can change or become less accurate as more data becomes available or conditions change.

Solving Algebraic Equations

Solving algebraic equations is a critical skill in mathematics that involves finding the value(s) of the variable(s) that make the equation true. The steps typically include simplifying both sides of the equation, isolating the variable, and making any necessary computations to solve for that variable.

For instance, to find when the US population reached 50 million, we solved the equation \(50,000,000 = 5,500,400 + 683,300t^{2}\) for \(t\). It started with isolating \(t^{2}\) and performing arithmetic operations such as subtraction and division. After calculating \(t^{2}\), we found \(t\) by taking the square root. Understanding each step in solving algebraic equations helps ensure accuracy and provides a foundation for tackling more complex mathematical problems.