Problem 26 - Algebra A Combined Function

Question

The penetration rate of American wireless telephone subscribers \(-\) that is, the percent of the population who have cell phones \(-x\) years after 1995 is given by \(0.1 x^{2}+4.4 x+10.7\) for 1995 through \(2008 .\) Assuming the same rate of growth, use this model to predict the penetration rate of wireless subscribers in the United States in \(2010(x=15) .\) (Source: Based on data from Cellular Telecommunications \& Internet Association)

Step-by-Step Solution

Answer

The predicted penetration rate in 2010 is 99.2%.

Step 1: Understand the Problem

We are given a quadratic function that models the penetration rate of wireless subscribers. The function is \(0.1x^2 + 4.4x + 10.7\). Here, \(x\) is the number of years after 1995. We need to find the penetration rate for the year 2010, which corresponds to \(x = 15\).

Step 2: Substitute \(x = 15\) in the Formula

Insert \(x = 15\) in the quadratic formula to find the penetration rate in 2010. The formula becomes: \(0.1(15)^2 + 4.4(15) + 10.7\).

Step 3: Calculate \(0.1(15)^2\)

Calculate \(15\) squared to get \(225\), and then multiply by \(0.1\) to get \(22.5\).

Step 4: Calculate \(4.4(15)\)

Multiply \(4.4\) by \(15\) to get \(66\).

Step 5: Sum All the Components

Add the results from the previous steps: \(22.5 + 66 + 10.7\) to get the total penetration rate for 2010.

Step 6: Final Calculation and Result

The sum \(22.5 + 66 + 10.7\) gives \(99.2\). Therefore, the predicted penetration rate in 2010 is \(99.2\%\).

Key Concepts

Penetration Rate

The penetration rate in the context of wireless subscribers refers to the percentage of people in a specific population who have adopted and utilize wireless telephone services. This metric provides insights into how widespread the use of technology is within a given market. A high penetration rate indicates a large majority of the population uses the service, while a low rate suggests there is potential for further growth.
In the exercise, we see penetration rate expressed as a function, aimed at forecasting how many people have adopted cell phones at different times post-1995. This helps in understanding market trends and potential saturation points for the mobile industry. By understanding this percentage, companies can tailor their offerings and marketing strategies to either state of high or low market penetration.

Predictive Modeling

Predictive modeling refers to the use of statistical techniques to create a model that can predict future outcomes based upon historical data. In this exercise, a quadratic function is utilized to predict the penetration rate of wireless subscribers. Predictive models like this are essential for companies in planning and strategizing future operations.
**How Predictive Modeling is Useful:**

• It helps in anticipating changes in market dynamics.
• Provides evidence-based projections which can guide decision-making.
• Facilitates understanding of complex relationships between variables.

Predictors generate forecasts by extending past patterns into the future, assuming those patterns continue to hold. This is what happens when we apply the year 2010 to the given function to predict future penetration rates.

Wireless Subscribers

Wireless subscribers are individuals who have a subscription or access to a wireless telephone service. In this context, these subscribers refer to the cell phone users in America, from 1995 onwards.
It is important to recognize that wireless subscribers have grown substantially since the mid-1990s due to technological advancements and increased affordability of devices and services. The data used here reflects these increases and shows how a mathematical function can demonstrate these changes over time.
In real-world applications, wireless subscriber data is crucial for network service providers to determine coverage areas, adjust service capacities, and plan for infrastructural development.

Function Evaluation

Function evaluation involves computing the output of a mathematical function for a given input. Specifically, in this practice exercise, we are evaluating a quadratic function to find the penetration rate in a future year (2010) by substituting specific values.
**Steps in Function Evaluation:**

• Identify the function: Understand the formula or equation you're working with, which here is the quadratic function.
• Substitute the variable: Plug in the given value, like substituting 15 for x when calculating the penetration for 2010.
• Perform calculations: Follow mathematical operations like squaring numbers, multiplying terms, and adding constants.
• Sum up all components to get the final result.

Function evaluation is a core part of predictive modeling, enabling the calculation of expected outcomes based on a set mathematical relationship.