Problem 120 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The model \(P=-0.18 n+2.1\) describes the number of pay phones, \(P,\) in millions, \(n\) years after \(2000,\) so I have to solve a linear equation to determine the number of pay phones in 2002

Step-by-Step Solution

Answer

The statement makes sense in terms of needing to use the model to find the number of pay phones in 2002, but it does not make sense in terms of having to 'solve a linear equation'. What is actually done is substituting a value into the model, rather than solving an equation.

Step 1: Understanding the Given Model

Examine the given model \(P=-0.18 n+2.1\). This is a linear equation where \(P\) is the number of pay phones in millions, and \(n\) is the number of years passing after the year 2000. Thus, for any given year after 2000, an appropriate value for \(n\) can be substituted into the equation to determine \(P\).

Step 2: Calculating the Year 2002

To find the number of pay phones in 2002 using the model, replace \(n\) with \((2002-2000) = 2\). So, the calculation to make is \(P=-0.18 (2)+2.1 = 1.74\). So, in 2002, the model predicts there are approximately 1.74 million pay phones.

Step 3: Assessing the Statement

The statement said that a linear equation needed to be solved to find the number of pay phones in 2002. However, what is really being done is substituting a particular value of 'n' (the years after 2000) into the equation to find the corresponding value of 'P' (the numbers of pay phone). Technically, it's not 'solving a linear equation', but is stucking a numerical value into a linear function, which is different.

Key Concepts

Substitution Method

The substitution method is a key concept when working with linear equations, especially in scenarios involving models. Here, we have a linear model given by the equation \(P = -0.18n + 2.1\). This equation shows the relationship between two variables: \(P\), the number of pay phones in millions, and \(n\), the number of years after 2000. To make use of this model, we use substitution.In this context, substitution involves replacing \(n\) with a specific value to calculate \(P\). For instance, to find the number of pay phones in 2002, you substitute \(n\) with \(2\) (since 2002 is 2 years after 2000). Thus, you substitute \(n = 2\) into the equation, giving you \(P = -0.18 \times 2 + 2.1\). By solving this, we find \(P = 1.74\). This demonstrates how substitution converts a linear model into a useful tool for predicting outcomes.

• Always identify the substitute variable.
• Replace the variable in the equation accordingly.
• Solve for the unknown value after substitution.

Substitution can simplify complex problems by focusing on one variable at a time.

Linear Models

Linear models are equations that express a linear relationship between two variables. In our case, the model \(P = -0.18n + 2.1\) represents such a relationship. The goal of a linear model is to predict one quantity based on another. Here, \(n\) helps us predict \(P\), the number of pay phones.This linear model is key to understanding how pay phone numbers change over the years.

The equation \(P = -0.18n + 2.1\) can be broken down as follows:

• Intercept (2.1): This is the starting value or predicted number of pay phones (in millions) at \(n = 0\), which is the year 2000.
• Slope (-0.18): This number shows the rate of change per year. Specifically, it tells us that the number of pay phones decreases by 0.18 million for every additional year after 2000.

Using linear models helps us project future outcomes and understand trends based on observable patterns.

Interpretation of Models

Interpreting a linear model involves understanding and explaining what the components of the model signify in practical terms. The model \(P = -0.18n + 2.1\) provides insights beyond mere calculation. Let's break down how to interpret this information effectively.

• The Slope: The slope \(-0.18\) is crucial as it reflects how fast the number of pay phones is declining each year. A negative slope indicates a decrease, while a positive one would indicate growth.
• The Intercept: The intercept \(2.1\) is the estimated number of pay phones at the beginning of the reference period (the year 2000).

Understanding what each element of the model represents helps in making real-world predictions and decisions.For example, by interpreting these values, one recognizes that according to the model, there will be fewer pay phones available each year after 2000, and this trend continues consistently into the future unless other factors intervene. Thus, interpreting models not only involves calculating values but also considering what these values mean conceptually.