Problem 47 - 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. Find the ratio in simplest form of Population 1 to Population 2, that is \(\frac{P_{1}}{P_{2}}.\)

Step-by-Step Solution

Answer

After analyzing and simplifying, the ratio \(\frac{P_{1}}{P_{2}} = \frac{x+3}{x+1}\) , where \(x\) represents years.

Step 1: Set up the ratio

Set up the ratio of the two populations, \(P_{1}\) and \(P_{2}\), as a fraction: \(\frac{P_{1}}{P_{2}}\). Substituting the given equations for \(P_{1}\) and \(P_{2}\), \(\frac{P_{1}}{P_{2}} = \frac{\frac{100 x^2}{x+1}}{\frac{100 x^2}{x+3}} \).

Step 2: Simplify the ratio

Simplify the division of fractions by multiplying the numerator by the reciprocal of the denominator: \(\frac{P_{1}}{P_{2}} = \frac{100x^{2}}{x+1} * \frac{x+3}{100x^{2}} \).

Step 3: Reduce common terms

Cancel common terms in the numerator and denominator. The \(100x^{2}\) in the numerator and denominator cancel each other out. This leaves \(\frac{P_{1}}{P_{2}} = \frac{x+3}{x+1}\).

Key Concepts

Simplifying Algebraic Expressions

An essential skill in algebra is simplifying algebraic expressions, which essentially means to reduce them to their simplest form. In the context of the given exercise, simplifying involves several steps that can be applied to most algebraic expressions.

One begins by setting up the correct expression for simplification. In this case, the ratio of two populations is the focus, leading to an initial complex fractional expression. To simplify, it's necessary to look for common factors that can be eliminated. Factors that appear in both the numerator and the denominator, like the term \(100x^{2}\) in our exercise, can be cancelled out. This cancellation is grounded in one of the fundamental properties of fractions, that a fraction does not change its value when both the numerator and the denominator are multiplied or divided by the same non-zero number.

General Steps for Simplification

• Identify and eliminate common factors.
• Perform arithmetic operations such as multiplication or division as needed.
• Combine like terms and reduce expressions to the fewest terms possible.

These steps not only make the expression easier to work with but also help to gain a clearer understanding of what the expression represents. Simplifying algebraic expressions aids in problem-solving and allows for better comparison when dealing with ratios, as seen with the populations \(P_{1}\) and \(P_{2}\).

Rational Expressions

Rational expressions are fractions that include polynomials in their numerator, denominator, or both. They are the ratio of two algebraic expressions and require careful manipulation when simplifying, as seen in our textbook exercise. To handle these expressions correctly, one should be familiar with the rules for simplifyings fractions and the operations with polynomials.

Rational expressions can be simplified by reducing common factors. In the exercise, we simplified a complex rational expression, representing population ratios, by applying the division rule for fractions—multiplying by the reciprocal. This technique is crucial for simplifying rational expressions, as it transforms division into multiplication, often making it easier to spot and cancel out common terms.

The final simplified form of the rational expression provides the simplest way to compare the two mentioned populations, giving clearer insights into their relative sizes over time. Recognizing and reducing rational expressions is, therefore, a valuable skill in algebra and various applications of mathematics.

Mathematical Modeling

Mathematical modeling is a powerful tool for representing real-world situations using mathematical concepts and language. In our given exercise, the populations of two prairie dog colonies are modeled by algebraic expressions, with time as the independent variable.

This practice allows scientists and mathematicians to analyze complex phenomena through a mathematical lens, making it possible to predict future behaviors or compare different scenarios. In the case of the prairie dog populations, models in the form of rational expressions provide insight into how the populations change over time and how they relate to each other.

Importance of Accurate Modeling

• Models can predict behaviors, trends, and outcomes.
• They aid in understanding the relationship between variables.
• Models can be refined and improved with new data.

A robust mathematical model can lead to better decision-making and deeper insights into the studied phenomena. Simplifying expressions within these models, as we did with the populations' ratio, allows us to interpret and compare outcomes more efficiently.