Problem 25 - Algebra A Combined Function

Question

The number of wireless telephone subscribers (in millions) \(x\) years after 1995 is given by the polynomial \(0.52 x^{2}+11.4 x+27.87\) for 1995 through 2008\. Use this model to predict the number of wireless telephone subscribers in \(2012(x=17)\) (Source: Based on data from Cellular Telecommunications \& Internet Association)

Step-by-Step Solution

Answer

371.95 million subscribers in 2012.

Step 1: Understand the Problem

We are given a polynomial function that represents the number of wireless telephone subscribers (in millions) from the year 1995 to 2008. The function is \( f(x) = 0.52x^2 + 11.4x + 27.87 \), where \( x \) is the number of years after 1995. We need to use this model to predict the number of subscribers in 2012, which corresponds to \( x = 17 \), since 2012 is 17 years after 1995.

Step 2: Substitute x=17 into the Polynomial Function

Substitute \( x = 17 \) into the equation \( f(x) = 0.52x^2 + 11.4x + 27.87 \). This allows us to calculate the number of subscribers in the year 2012: \[ f(17) = 0.52(17)^2 + 11.4(17) + 27.87 \].

Step 3: Calculate the Squared Term

First, calculate \( 17^2 \), which is the squared term: \[ 17^2 = 289 \]. Then substitute this into the equation: \[ 0.52 \times 289 \].

Step 4: Multiply and Simplify

Calculate \( 0.52 \times 289 = 150.28 \). Next, calculate \( 11.4 \times 17 = 193.8 \). Substitute these results back into the function: \[ f(17) = 150.28 + 193.8 + 27.87 \].

Step 5: Add the Results

Add the calculated numbers together: \[ 150.28 + 193.8 + 27.87 = 371.95 \].

Step 6: Conclusion

The number of wireless telephone subscribers predicted for the year 2012 is 371.95 million, according to the given polynomial model.

Key Concepts

Understanding the Substitution Method

In mathematics, the substitution method is a powerful tool for solving equations and predicting values in mathematical models. The core idea is to replace a variable with a specific number to find the outcome of a function. This technique is often used in polynomial functions.

Here's a quick breakdown of how to apply it:

• Select the equation you need to solve or evaluate. In our exercise, the polynomial function given is \( f(x) = 0.52x^2 + 11.4x + 27.87 \).
• Determine the value for the variable. In this problem, we need to predict for the year 2012, which means \( x = 17 \), because 2012 is 17 years after 1995.
• Substitute the selected value into the equation. Replace \( x \) with 17 in the function to calculate \( f(17) \).
• Simplify to solve. Calculate each part, as shown in steps, to find the final result. This involves arithmetic operations like squaring the number, multiplying, and adding.

The substitution method not only finds the specific values required but also verifies the suitability of polynomial models for predictions over certain intervals.

The Role of Mathematical Modeling

Mathematical modeling serves as a framework for representing real-world phenomena through mathematical expressions and functions. This exercise illustrates how a polynomial function models the growth in wireless telephone subscribers from 1995 to 2008.

Why use mathematical models like a polynomial?

• They provide a simplified representation of complex systems.
• Models allow us to understand trends and forecast future events using historical data.
• They turn abstract concepts into concrete numbers, which can be analyzed and interpreted.

In our exercise, the polynomial \( 0.52x^2 + 11.4x + 27.87 \) was created using past subscription data. This model allows us to project future subscription numbers beyond the original time frame by applying the substitution method. Mathematical modeling thus plays a crucial role in data interpretation and decision-making processes.

Predictive Analysis with Polynomial Functions

Predictive analysis is an application of mathematical tools to project future events and conditions based on historical data. Using polynomial functions for predictive analysis provides an understanding of potential trends and outcomes.

Here's how it works in the context of the exercise:

• We start with a polynomial function derived from historical subscriber data. This function captures the underlying trend over the years.
• By applying the substitution method, we can compute projections or predictions for years beyond the data range, such as 2012 in this scenario.
• Predictions can help businesses make informed strategic decisions by anticipating future demands or changes.

In practice, predictive analysis isn't just about the numbers; it's about leveraging mathematical models to gain insights and prepare for future possibilities. Polynomial functions, due to their flexibility in fitting various types of data, are particularly useful in making forecasts in fields like telecommunications. By understanding these models, professionals can not only predict but also adapt to approaching trends.