Problem 44 - Algebra A Combined Function

Question

The equation \(y=136.76(1.115)^{x}\) gives the number of cellular phone users y (in millions) in the United States for the years 2002 through \(2009 .\) In this equation \(x=0\) corresponds to \(2002, x=1\) corresponds to \(2003,\) and so on. Use this model to solve Exercises 43 and \(44 .\) Round answers to the nearest tenth of a million. Predict the number of cell phone users in 2014 .

Step-by-Step Solution

Answer

Approximately 504.6 million users in 2014.

Step 1: Identify the Year Corresponding to x

Since the model defines \(x=0\) as the year 2002, we need to calculate the value of \(x\) for the year 2014.\ Thus, \(x = 2014 - 2002 = 12\).

Step 2: Substitute x into the Equation

We substitute \(x = 12\) into the given equation to find the number of users.\[ y = 136.76(1.115)^{12} \]

Step 3: Calculate the Exponential Part

Calculate the value of \((1.115)^{12}\) first.\[ (1.115)^{12} \approx 3.689 \]

Step 4: Calculate y

Substitute the value from Step 3 into the equation from Step 2:\[ y = 136.76 \times 3.689 \]Calculate the product to find the approximate value.

Step 5: Final Calculation and Rounding

Perform the multiplication:\[ y = 504.56 \]Round 504.56 to the nearest tenth as the final answer yields 504.6.

Key Concepts

Mathematical Modeling

Mathematical modeling is an essential process for understanding and predicting real-world behaviors. It involves using mathematical expressions to represent complex phenomena. In the context of the original exercise, mathematical modeling is applied to estimate the number of cellular phone users over time. By using a model, we can create an equation to predict future trends based on past and present data.

Some benefits of mathematical modeling include:

• Ability to simulate scenarios and predict future outcomes.
• Understanding relationships between variables.
• Making decision-making processes more efficient.

In our example, the equation \(y=136.76(1.115)^{x}\) symbolizes the mathematical model. It is designed to project the growth trend of mobile users from the year 2002 onward. Each component of this formula (like the base 1.115 and the initial value of 136.76) holds significant meaning in this predictive exercise.

Exponential Functions

Exponential functions are a crucial part of mathematical modeling, especially when they involve topics like growth and decay. These functions grow by consistent percentages over equal increments of time, which is why they are often used in predictive analytics for rapid growth scenarios.

The core form of an exponential function is \(f(x) = a(b)^x\), where:

• \(a\) is the initial amount—or starting point,
• \(b\) characterizes the growth rate,
• \(x\) is the time, or number of periods.

The exercise from the textbook uses the specific exponential function \(y=136.76(1.115)^{x}\), demonstrating how the model anticipates rapid growth in a technological context. The base \(1.115\) serves as the multiplier that indicates a 11.5% increase annually. The exponential function is powerful because it can depict trends over time that start slow but can quickly escalate or decrease, depending on the context.

Predictive Modeling

Predictive modeling is the process of using known data to anticipate future outcomes. In our exercise, the model helps forecast the number of cell phone users in the United States for a given future year, such as 2014.

Core elements of predictive modeling include:

• Defining the problem and gathering data from past periods.
• Choosing the correct type of model (like linear or exponential).
• Validating the model to ensure accuracy with historical data before predicting future scenarios.

In our example, predictive modeling was used by calculating the number of years between 2002 and 2014 (\(x=12\)), and substituting it into the mathematical model \(y = 136.76(1.115)^{x}\). By doing this, we achieved an approximate prediction of 504.6 million users for the year 2014. Predictive modeling offers invaluable insights for businesses, researchers, and policymakers, enabling them to plan strategically based on anticipated future trends.