Problem 116 - 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. Predict the number of cellular telephone subscriptions in \(2015 .\) Round to the nearest tenth of a million.

Step-by-Step Solution

Answer

The predicted number of subscriptions in 2015 is 230.0 million.

Step 1: Identify the value of x for 2015

The function \( f(x) \) models the number of cellular telephone subscriptions in millions, \( x \) years after 1995. To find \( x \) for the year 2015, subtract 1995 from 2015: \[ x = 2015 - 1995 = 20 \]

Step 2: Substitute x into the function

Substitute \( x = 20 \) into the function \( f(x) = 25 x^{23/25} \): \[ f(20) = 25 \times 20^{23/25} \]

Step 3: Calculate the exponentiation

First, calculate the exponentiation \( 20^{23/25} \). Using a calculator: \[ 20^{23/25} \approx 9.198 \]

Step 4: Multiply the results

Now, multiply the result from the exponentiation by 25 to find the number of subscriptions: \[ f(20) = 25 \times 9.198 = 229.95 \]

Step 5: Round to the nearest tenth

Round the computed value \( 229.95 \) to the nearest tenth: \[ 229.95 \approx 230.0 \] Therefore, the predicted number of cellular telephone subscriptions in 2015 is approximately 230.0 million.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base, and the exponent. It's represented as \( x^n \), where \( x \) is the base, and \( n \) is the exponent. This operation means multiplying the base \( x \) by itself \( n \) times. For fractional exponents, we take the nth root of the base. This concept is vital in mathematical modeling, like predicting future trends or behaviors, such as the number of cell phone subscribers in a future year.

In our example, the model function is \( f(x) = 25 x^{23/25} \), where the exponent \( 23/25 \) is fractional. This tells us that we're calculating a root-like result based on the given base. Calculating \( 20^{23/25} \) reveals an approximated value which models the rise in subscriber numbers efficiently over the years.

Problem Solving

Problem solving is an essential skill in mathematics and everyday life. It's about identifying the issue, determining ways to address it, and testing potential solutions until you find the most efficient one.

In our exercise, the problem is to predict future cellular subscriptions using a formula. Here's a systematic approach:

• Understand the problem: Recognize the year and find the count of years from the given starting point.
• Formulate a strategy: Identify the formula provided and prepare to substitute the correct variable.
• Solve: Implement the calculation by evaluating the formula with specific values.
• Verify: Check if the calculated result makes sense, often by rounding for practicality.

Approaching problems with structured thinking ensures better and more accurate outcomes.

Function Evaluation

Function evaluation is a process by which we determine the output of a function based on given inputs. A function is generally a mathematical relationship expressed as \( f(x) \). It produces an output when we substitute a specific value in place of \( x \).

For instance, in the exercise \( f(x) = 25 x^{23/25} \), we're tasked with finding the number of subscriptions in 2015. We start by identifying \( x \) as the number of years after 1995, which calculates to 20 in this scenario. Substituting 20 into \( f(x) \) gives us the function evaluation \( f(20) = 25 \times 20^{23/25} \). Solving these computations is crucial in making accurate predictions, often encountered in data analysis and forecasting tasks.

Rounding Numbers

Rounding numbers simplifies numerical values by reducing the number of digits, making them easier to interpret or approximate for practical purposes. It's a common practice, particularly in reporting or when an exact figure isn't necessary.

In mathematics, when instructed to round to the nearest tenth, focus on the first decimal number. If the following decimal is 5 or greater, round up. If less than 5, round down. In the problem, we calculate a figure of 229.95 million telephone subscriptions. Rounding 229.95 to the nearest tenth results in 230.0. This helps in presenting a clearer, more digestible result to audiences without sacrificing much accuracy. Use rounding to facilitate easier reading and analysis in practical applications.