Problem 61 - Algebra 1: Concepts and Skills

Question

POPULATION The population \(P\) of Texas (in thousands), as projected through \(2025,\) is modeled by \(P=18,870(1.0124)^{t},\) where \(t=0\) represents \(1995 .\) Find the ratio of the population in 2025 to the population in 2000 .

Step-by-Step Solution

Answer

The ratio of the population in 2025 to the population in 2000 is given by \((1.0124)^{25}\).

Step 1: Calculate Population in 2000

To find the population in 2000, we need to substitute \(t = 5\) (since 2000 corresponds to \(t = 5\) when (\(t = 0\) represents 1995) into the equation for \(P\). This gives \(P_{2000} = 18,870(1.0124)^5\).

Step 2: Calculate Population in 2025

Next, to find the population in 2025, substitute \(t = 30\) (since 2025 corresponds to \(t = 30\) when (\(t = 0\) represents 1995) into the equation for \(P\). This gives \(P_{2025} = 18,870(1.0124)^{30}\).

Step 3: Calculate Ratio

Lastly, to find the ratio of the population in 2025 to the population in 2000, we divide \(P_{2025}\) by \(P_{2000}\), giving \(\frac{P_{2025}}{P_{2000}} = \frac{18,870(1.0124)^{30}}{18,870(1.0124)^5}\).

Step 4: Simplify Ratio

Since the population in 2000 and 2025 share the same base, the ratio simplifies to \(\frac{P_{2025}}{P_{2000}} = (1.0124)^{25}\).

Key Concepts

Population Growth

Population growth is a fascinating and significant aspect of exponential functions. In this example, the population of Texas is modeled over a 30-year period using the formula \( P = 18,870(1.0124)^t \). Here, changes in population size over time can be analyzed efficiently. The number \( 18,870 \) represents the initial population at the start year, 1995. The general formula for population growth shows how populations increase over time, influenced by factors like birth rates, death rates, and migration.
Unlike linear growth, where the increase is constant each year, exponential growth means each year's population depends on the previous year's population, resulting in faster growth over time. This is shown through the base number \( 1.0124 \) in the equation, which adds a fractional growth rate of 1.24% per year.
Understanding population growth through these formulas helps in planning and resource management, as it predicts future needs based on current trends.

Algebraic Modeling

Algebraic modeling involves using algebraic equations to represent real-world scenarios. In our population growth model, the equation \( P = 18,870(1.0124)^t \) represents the population over time. Such models are invaluable because

• They provide a mathematical representation of a phenomenon.
• They allow for the prediction of future values based on observed data trends.
• They can be adjusted for changes in growth conditions, enhancing flexibility.

In this context, algebraic modeling condenses complex population changes into a simple formula that can be used for repeated calculations. This makes it easier to predict populations at future dates, such as 2000 and 2025, by substituting different values of \( t \) into the formula.
The power of algebraic modeling lies in its ability to take intricate systems and simplify them into manageable pieces, making them accessible for analysis and predictions.

Ratio Calculation

Calculating ratios is a key method in comparing different quantities, allowing us to understand the relationship between them. In this exercise, after calculating the populations in 2000 and 2025, the objective is to find the ratio of these two populations. This ratio effectively tells us how many times larger or smaller the population is at one point compared to another.
To find this, we simplify the division of the population in 2025 by the population in 2000: \[ \frac{P_{2025}}{P_{2000}} = \frac{18,870(1.0124)^{30}}{18,870(1.0124)^5} \] On simplifying, both \( 18,870 \) terms cancel out, leaving us with: \( (1.0124)^{25} \). This shows the compounded growth factor of the population over the 25-year period.
Understanding ratios lets us compare different quantities efficiently without needing to know the actual figures, offering a clear picture of growth trends and comparisons over time.