Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

The table lists the total numbers of radio stations in the United States for certain years. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Number } \\ \hline 1950 & 2773 \\ \hline 1960 & 4133 \\ \hline 1970 & 6760 \\ \hline 1980 & 8566 \\ \hline 1990 & 10,770 \\ \hline 2000 & 12,717 \\ \hline \end{array} $$ (a) Determine a linear function \(f(x)=a x+b\) that models these data, where \(x\) is the year. (b) Find \(f^{-1}(x)\). Explain the significance of \(f^{-1}\). (c) Use \(f^{-1}\) to predict the year in which there were 11,987 radio stations. Compare it with the true value, which is 1995 .

Step-by-Step Solution

Answer

The predicted year using the model is 1996, close to the true value of 1995.

Step 1: Understanding the Request

We need to find a linear function \( f(x) = ax + b \) that models the data of radio stations over the years. This involves using the method of least squares to find \( a \) and \( b \). Additionally, we need the inverse of this function \( f^{-1}(x) \) to predict which year had a given number of radio stations. We later need to compare our prediction to an actual known value from 1995.

Step 2: Calculating the Slope (a)

The slope \( a \) of the line \( f(x) = ax + b \) can be determined using the formula for the slope of a line: \[ a = \frac{y_2 - y_1}{x_2 - x_1} \]Let's choose the points for 1950 (2773) and 2000 (12717) as endpoints:\[ a = \frac{12717 - 2773}{2000 - 1950} = \frac{9944}{50} = 198.88 \] (rounded to two decimal places).

Step 3: Calculating the Intercept (b)

To find the intercept \( b \), use the slope-point formula \( y = ax + b \) with one of the data points (for instance, 1950, 2773):\[ 2773 = 198.88 \times 1950 + b \]\[ b = 2773 - (198.88 \times 1950) \]\[ b = 2773 - 387816 = -384943 \] (rounded to an integer).

Step 4: Constructing the Function

We now have the slope \( a = 198.88 \) and intercept \( b = -384943 \). So, the linear function is:\[ f(x) = 198.88x - 384943 \]

Step 5: Finding the Inverse Function

To find the inverse function \( f^{-1}(x) \), solve for \( x \) in terms of \( y \):\[ y = 198.88x - 384943 \]\[ x = \frac{y + 384943}{198.88} \]Thus, the inverse is:\[ f^{-1}(x) = \frac{x + 384943}{198.88} \]

Step 6: Using Inverse to Predict the Year

Use the inverse function \( f^{-1}(x) \) to find the year when there were 11,987 radio stations:\[ f^{-1}(11987) = \frac{11987 + 384943}{198.88} \]\[ = \frac{396930}{198.88} \approx 1995.64 \]The predicted year, when rounded, is 1996.

Step 7: Comparing Prediction with True Value

The true value given was 1995. Our prediction of 1996 is quite close, indicating that the model is reasonably accurate in this range.

Key Concepts

Inverse Function

An inverse function essentially reverses the operation of the original function. Given a function that maps an input to an output, the inverse function does the opposite.In our context, the original function models the number of radio stations as a linear function of the year. The inverse tells us the year associated with a specific number of radio stations. This is especially helpful for predicting historical data trends.
To find the inverse, we switch the roles of the dependent and independent variables. Start with the function equation and solve for the original independent variable. For a linear function of the form:

• \( y = ax + b \)
• Switch to \( x = \frac{y - b}{a} \)

Thus, this new equation helps us track back from given data like the number of stations, to derive the corresponding year.
The importance of understanding inverse functions expands beyond this problem. It's a crucial concept in many mathematical applications, including transformations and real-world changes.

Data Modeling

Data modeling is about creating a mathematical description that represents real-world data. In the exercise, we are modeling the number of radio stations using a linear equation. The goal is to find the best-fit line that can represent the trend seen in the data points provided. For linear regression, the objective is to determine the slope and intercept values that minimize the difference between observed data and predicted results (the least squares method). Data modeling requires:

• Choosing appropriate data points for calculating slope and intercept.
• Building a model (equation) based on calculated values.
• Interpreting and validating the outcomes using existing or predicted data.

Through these steps, we can derive meaningful insights and make predictions, as shown when we predict the year based on the stations count. This form of modeling is powerful in examining historical patterns and forecasting future trends.

Slope-Intercept Form

The slope-intercept form is a linear equation format used widely in algebra. It makes equations simple to read and understand, showing the slope (rate of change) and the y-intercept (the starting point when \(x=0\)).The general expression is:

• \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

For our function, the slope \( m \) indicates how many radio stations are added per year, while \( c \) shows where the line crosses the y-axis when there are hypothetically no years counted, providing insight into the linear pattern present within the data.
Understanding each element is important:

• Slope \( m \): Represents the steepness of the line. More stations with time show a positive slope.
• Intercept \( c \): Helps predict starting conditions within the context of the model.

We utilized these components to identify the best-fitting line from the radio station data, making our model useful for predicting outcomes.