Problem 48 - Algebra 1: Concepts and Skills

Question

Use the following information.Scientists are monitoring two distinct prairie dog populations,\(P_{1}\) and \(P_{2},\)modeled as follows. \(P_{1}=\frac{100 x^{2}}{x+1}\) and \(P_{2}=\frac{100 x^{2}}{x+3}\) where x is time in years. Add another row to your table labeled \(\frac{P_{1}}{P_{2}}\) and evaluate for each value of x.

Step-by-Step Solution

Answer

The ratio \(P_{1}/P_{2}\) is simplified to \(R = \frac{x+3}{x+1}\). The specific values for \(x\) would be substituted into this formula to calculate the respective ratios.

Step 1: Define the Ratio

The first step is to define the ratio, which we will symbolize as \(R\). The task requires the ratio of \(P_{1}\) to \(P_{2}\), which can be written as \(R = \frac{P_{1}}{P_{2}}\), where \(P_{1}\) is the population model for the first prairie dog population and \(P_{2}\) is the population model for the second prairie dog population.

Step 2: Substitute the formulas of \(P_{1}\) and \(P_{2}\) into the ratio

In this step, the formulas given for \(P_{1}\) and \(P_{2}\) will be substituted into the ratio: \(R = \frac {\frac{100x^{2}}{x+1}}{\frac{100x^{2}}{x+3}}\)

Step 3: Simplifying the Ratio

The next step is to simplify the formula for \(R\). Using the property that dividing fractions is the same as multiplying by the reciprocal, the ratio simplifies to: \(R = \frac{\frac{100x^{2}}{x+1}}{\frac{100x^{2}}{x+3}} = \frac{100x^{2}}{x+1} \cdot \frac{x+3}{100x^{2}} = \frac{x+3}{x+1}\)

Step 4: Evaluate the Ratio

Finally, values for \(x\) are plugged into the function \(R(x) = \frac{x+3}{x+1}\) to find their corresponding \(R\) values. The exact x values to be used depend on the context and the instructions of the problem.

Key Concepts

Population Models

Population models are tools used by scientists to understand how a population changes over time. In our exercise, the prairie dog populations are modeled using mathematical expressions: \(P_1 = \frac{100x^2}{x+1}\) and \(P_2 = \frac{100x^2}{x+3}\). These expressions describe how each population grows or declines as time passes, with \(x\) representing time in years.
These models are crucial in studying ecological populations as they help predict future changes and help in conservation efforts.

• Factors in Population Models: Typically include birth rates, death rates, and migration patterns, but here, we focus on changes over time.
• Understanding Growth: The expressions show how the populations scale with time using parameters specific to each group.

By comparing population models, such as \(P_1\) and \(P_2\), scientists can assess differences in growth rates due to different environmental conditions or genetic differences between the prairie dog populations.

Simplifying Fractions

Simplifying fractions in mathematical expressions can make them easier to work with and understand. The simplification of fractions involves rewriting them in their most reduced form.
In this exercise, the ratio \(R = \frac{\frac{100x^2}{x+1}}{\frac{100x^2}{x+3}}\) was simplified.
Let's break it down:

• Fraction Division: Dividing fractions is equivalent to multiplying by the reciprocal. So, \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \).
• Simplification Process: For the given ratio, it becomes \(R = \frac{100x^2}{x+1} \times \frac{x+3}{100x^2} = \frac{x+3}{x+1}\).

Through these steps, the complexity of the original expression is reduced, making it easier to analyze and evaluate.

Evaluating Functions

Evaluating functions is the process of finding the output of a function for specific input values. It helps to understand how the function behaves under different conditions. In our context, we evaluate the function of the simplified ratio \(R(x) = \frac{x+3}{x+1}\) for different \(x\) values.

• Input Substitution: Substitute a specific value of \(x\) into the function.
• Calculate: Perform arithmetic operations to find the value of \(R(x)\).

By doing this for several values of \(x\), students can observe how the ratio changes over time, providing insights into comparative growth trends of the prairie dog populations.
This practice not only aids in understanding the function's behavior but also enhances analytical skills in mathematical modeling.