Problem 116 - A Graphical Approach to Precalculus with Limits

Question

Vehicle Sales In 2009 new motor vehicle sales in the United States were \(10,602\) thousand. In 2013 the figure had increased to \(15,844\) thousand. (a) Find a linear function \(P(x)=a x+b\) that models the number of vehicle sales \(x\) years after 2009 . (b) Interpret the slope of the graph of \(y=P(x)\) (c) Use \(P(x)\) to approximate the number of vehicle sales in 2011 (d) Assuming the model continued past 2013 , what would be the number of sales in \(2015 ?\)

Step-by-Step Solution

Answer

(a) \(P(x) = 1310.5x + 10602\); (b) Slope is 1310.5 thousand sales/year; (c) 13,223 thousand sales in 2011; (d) 18,465 thousand sales in 2015.

Step 1: Determine the slope (a)

The slope of the linear function is determined by the change in vehicle sales divided by the change in years. We have two points: (0, 10602) for 2009 and (4, 15844) for 2013. \[ a = \frac{y_2 - y_1}{x_2 - x_1} = \frac{15844 - 10602}{4 - 0} = \frac{5242}{4} = 1310.5 \] So the slope \( a = 1310.5 \).

Step 2: Determine the y-intercept (b)

Using the point (0, 10602) and the slope, we can find the y-intercept \( b \). The equation is: \[ b = y - ax \] Substituting the known values, we have: \[ b = 10602 - (1310.5 \times 0) = 10602 \] Hence \( b = 10602 \).

Step 3: Write the linear function P(x)

Now that we have both \( a \) and \( b \), the linear function is: \[ P(x) = 1310.5x + 10602 \] This function models the number of vehicle sales \( x \) years after 2009.

Step 4: Interpret the slope

The slope \( a = 1310.5 \) represents the rate of increase in vehicle sales per year. For each additional year, vehicle sales increased by an average of 1310.5 thousand units.

Step 5: Approximate sales in 2011

To find the approximate number of vehicle sales in 2011, substitute \( x = 2 \) into the function \( P(x) \): \[ P(2) = 1310.5 \times 2 + 10602 = 2621 + 10602 = 13223 \] So, the approximate sales in 2011 were 13,223 thousand vehicles.

Step 6: Predict sales in 2015

Assuming the trend continued, substitute \( x = 6 \) to predict sales in 2015: \[ P(6) = 1310.5 \times 6 + 10602 = 7863 + 10602 = 18465 \] The predicted number of sales in 2015 is 18,465 thousand.

Key Concepts

Mathematical Modeling

Mathematical modeling involves creating equations to describe real-world situations. In this exercise, we are using a linear function to model vehicle sales over time.
A linear function is a straight line equation represented in the form \(P(x) = ax + b\). Here, \(x\) is the number of years since a starting point (2009) and \(P(x)\) represents vehicle sales in thousands.
For our specific problem:

• The linear function that models our data is \(P(x) = 1310.5x + 10602\). This was determined by analyzing sales data from 2009 and 2013.
• The goal of this model is to predict sales at other points in time based on past trends. By doing so, we can make informed guesses about future sales, assuming the same trends continue.

Understand that mathematical models are approximations. They provide insights but are not exact reflections of reality. This function serves well as a basic tool for understanding sales growth.

Slope Interpretation

The slope of a linear function is an important concept. It provides information about the rate of change between two variables. In our exercise, the slope tells us how vehicle sales change over time.
In the function \(P(x) = 1310.5x + 10602\), the slope is \(1310.5\). This number means that every year, vehicle sales increase by approximately 1310.5 thousand units.
Key points to keep in mind:

• The slope is positive, indicating growth in sales over the years considered.
• A greater slope value signifies faster growth, whereas a smaller value suggests slower growth.
• Understanding the slope helps in identifying patterns and making predictions about future trends.

The slope helps answer questions such as "+1310.5 per year" effectively and provides a clear mathematical representation of change.

Linear Equations

Linear equations form the foundation of many mathematical and real-world applications. Recognizing and using linear equations allows us to solve problems efficiently.
A linear equation like \( P(x) = 1310.5x + 10602\) comprises two main components:

• The coefficient \(a = 1310.5\) is the slope. It represents how the dependent variable (sales) changes with the independent variable (year).
• The y-intercept \(b = 10602\) represents the starting point. It is the value of sales when \(x = 0\), which in our context corresponds to sales in 2009.

Linear equations are invaluable tools for interpolating and extrapolating data. They allow predictions between known points or estimating beyond given data, as in predicting vehicle sales for 2011 and 2015 in our case.
The simplicity of linear equations makes them accessible, yet their applications can be highly profound in understanding and forecasting trends.