Question

The population \(P\) of Texas (in thousands) for 1995 projected through 2025 can be 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 . (Review 8.3) P Source: U.S. Bureau of the Census.

Short answer

Calculate \(P_1 =18870(1.0124)^5\) to find the population in 2000 and \(P_2 = 18870(1.0124)^{30}\) to find the population in 2025. The ratio of these two populations is then \(\frac{P_2}{P_1}\)

Step-by-step solution

Substitute 't' for year 2000

To get the population in year 2000, substitute \(t = 2000 - 1995 = 5\) into the given model. The equation is now \(P_1 =18870(1.0124)^5. Calculate P_1\)

Substitute 't' for year 2025

To get the population for year 2025, substitute \(t = 2025 - 1995 = 30\) into the model. The equation is now \(P_2 = 18870(1.0124)^{30}. Calculate P_2.\)

Calculate the Ratio

Calculate the ratio by dividing the population in 2025 by the population in 2000. \(Ratio = \frac{P_2}{P_1}\)

Key concepts

Exponential Growth

Understanding exponential growth is critical when predicting behaviors that increase at a constant percentage over time, such as population numbers. In the context of populations, the exponential model will often take the form of \(P(t) = P_0e^{rt}\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population at time \(t = 0\), \(r\) is the rate of growth, and \(e\) is the base of the natural logarithm.

However, the given Texas population model, \(P=18,870(1.0124)^t\), is slightly different. Here, the number 1.0124 represents a growth factor per year, not the rate. Specifically, this implies that each year the population increases by 1.24% (\(1.0124 - 1 = 0.0124\), or 1.24%). This compound growth leads to dramatic increases over longer periods of time, due to the population at each step being multiplied by a factor greater than 1.

Mathematical Modeling

Mathematical modeling is a potent tool for representing real-world situations with mathematical formulas. It helps us understand, predict, and potentially control events. The Texas population model is an example of using a mathematical approach to project growth. To create this particular model, data from the past was analyzed to determine a consistent growth rate. Then, the exponential formula was applied to encapsulate this trend into an equation.

In the case of the Texas population, the model assumes that the growth rate is constant over time, which is a simplification that might not always align with real-world complexities. Nevertheless, such models are instrumental in urban planning, resource management, and policy making. Improvements to the model could include adjusting for changes in birth rates, migration patterns, and policies that might significantly impact population dynamics.

Algebraic Problem-Solving

Algebraic problem-solving aids in deciphering complex scenarios by breaking them down into smaller, manageable algebraic steps. To solve the given problem, we employ simple algebra to first find the population in two different years and then divide one by the other to get the ratio. Here's a breakdown of the process:

By substituting the respective years into the model, we determined the population figures for both 2000 and 2025. The sequential execution of these algebraic manipulations—substitution followed by exponentiation and then division—illustrates a methodical approach to solving problems algebraically. When approaching algebraic problem-solving, it's vital to identify what each part of the equation represents and how altering these parts affects the overall equation or solution.