Problem 115 - Intermediate Algebra

Question

The number of cell telephone subscribers in the United States from \(1995-2010\) can be modeled by \(f(x)=25 x^{23 / 25},\) where \(f(x)\) is the number of cellular telephone subscriptions in millions, \(x\) years after \(1995 . \text { (Source: CTIA-Wireless Association, } 1995-2010)\) Use this information to answer. Use this model to estimate the number of cellular subscriptions in \(2010 .\) Round to the nearest tenth of a million.

Step-by-Step Solution

Answer

308.8 million subscriptions in 2010.

Step 1: Determine the Value of x

To use the model, we need to determine the value of \(x\). Since \(x\) represents the number of years after 1995, for the year 2010, we calculate \(x = 2010 - 1995 = 15\).

Step 2: Substitute x into the Model

Substitute \(x = 15\) into the model \(f(x) = 25x^{23/25}\). This gives us \(f(15) = 25(15)^{23/25}\).

Step 3: Calculate the Exponent

Calculate \(15^{23/25}\). Using a calculator, \(15^{23/25} \approx 12.3532\).

Step 4: Multiply by Coefficient

Multiply the result from Step 3 by 25: \(25 \times 12.3532 \approx 308.83\).

Step 5: Round to the Nearest Tenth

Round the result from Step 4 to the nearest tenth: \(308.83\) rounds to \(308.8\) million.

Key Concepts

Understanding Exponential Functions in Intermediate Algebra

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. This type of function is fundamental in various fields such as finance, biology, and physics because it models growth and decay processes.
In this context, the exponential function is represented as \( f(x) = 25x^{23/25} \). Here, "25" is a constant that scales the function, and \( x^{23/25} \) shows the growth pattern over time. This specific form illustrates how cell phone subscriptions increased over years.

• The base in this scenario is "x," the number of years since 1995.
• The exponent "\( \frac{23}{25} \)" dictates the growth pattern, indicating a less-than-exponential growth rate over time.

Understanding how to manage exponential functions helps us predict future outcomes or decode past trends, like our exercise demands to predict subscription numbers.

The Importance of Rounding Numbers

Rounding numbers is an important skill in mathematics, especially when you deal with real-world data. In our exercise, once we calculated the number of subscribers, it was necessary to round the result to the nearest tenth.
Rounding simplifies figures by reducing the number of digits while retaining a number close to the original value. This is essential because it makes the number easier to express and understand.

• To round to the nearest tenth, we look at the digit in the hundredth place. If it is 5 or more, we round up.
• In the case of our example, the calculated value was 308.83, and rounding this to one decimal point results in 308.8 million.

Rounding is valuable in reporting data where precision is less crucial than clarity and simplicity.

Applying Mathematical Modeling to Predict Trends

Mathematical modeling is about using mathematical expressions to depict real-world phenomena and predict future events based on historical data. In this textbook exercise, the function \( f(x) = 25x^{23/25} \) is a model representing the growth trend in cell phone subscriptions over a specified period.
This model is built upon historical data from 1995 to 2010, offering insights into how the number of subscribers increased over 15 years.Key aspects to understand about mathematical modeling:

• It turns complex real-world situations into simplified mathematical expressions, helping identify patterns or trends.
• Mathematical models depend upon assumptions and can vary based on those assumptions, such as growth rates in our case function.

By understanding and applying models, students can gain better insights into various phenomena, making way for data-driven decision-making.

The Substitution Method in Equation Solving

The substitution method involves replacing a variable in an equation with a numeral or another expression. This method is particularly beneficial when evaluating functions or solving systems of equations.
In our exercise, the goal was to estimate the number of subscriptions in 2010; therefore, it was essential to substitute the correct value of \( x = 15 \), representing the years past 1995, into the model \( f(x) = 25x^{23/25} \).

• Begin by finding the correct value of the variable (\( x = 15 \)).
• Substitute this value into the function to solve for \( f(x) \).
• This transformation simplifies calculations and gives results specific to given conditions.

The substitution method is a helpful technique, allowing students to simplify equations and solve them step by step systematically.