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Mathematical Modeling SECTION 1.1

3 Learning Objectives 1 Identify real uses for symbolic, numeric, and graphical forms of mathematical models 2 Analyze mathematical models and use them to create and answer real-world questions

4 Decision-Factor Equation

5 In this section we give a broad overview of how mathematical models are used to represent real-world problems. In later chapters, we demonstrate the process for finding such a model for a given situation. What is a mathematical model? Why are they used? Who could benefit from a mathematical model?

6 Decision-Factor Equation In this section we give a broad overview of how mathematical models are used to represent real-world problems. In later chapters, we demonstrate the process for finding such a model for a given situation.

7 Decision-Factor Equation Mathematical models help us understand the nature of problem situations. They are often helpful in making predictions or solving problems in real-world situations. One type of mathematical model is the decision-factor equation. We will use a decision-factor equation to model the decision process for purchasing a pre-owned vehicle.

8 Decision-Factor Equation When buying a vehicle, we carefully consider important features of a car such as price, manufacturer, engine type, fuel economy, color, model year, and body style, just to name a few. Suppose we are planning to purchase a preowned Toyota Corolla. We have decided that the three features most important to us are mileage, price, and color.

9 Decision-Factor Equation Table 1.1 shows Corollas meeting our criteria offered by Autotrader.com on July 6, 2007 in the vicinity of Indian Orchard, Massachusetts. Table 1.1

10 Decision-Factor Equation We can use the data in this table to create a decision-factor equation that will produce a number called the decision factor that will help us decide which car to buy. The car that best fits our criteria will have the smallest decision factor.

11 Example 1 – A Decision-Factor Equation Create a decision-factor equation with mileage, price, and color as the criteria. Assume that mileage is most important followed by price then color. Solution: Mileage and price have numeric values; color does not. We need to assign numeric values to the color options red, grey, and silver to use in our equation. We choose red = 1, grey = 2, and silver = 3, making red our first choice, grey our second choice, and silver our third choice.

12 Example 1 – Solution One decision-factor equation for this situation is Using Car 1 from Table 1.1, we get Notice that the color number has a negligible effect on the decision factor. cont’d Table 1.1

13 Example 1 – Solution If we want the color to have a greater effect, we can modify the decision-factor equation. For example, We use this modified equation to calculate the decision factor for each of the six cars on the list, as shown in Table 1.2. cont’d Table 1.2

14 Example 1 – Solution Using this model, we find that Car 6 has the lowest decision factor. Notice that it has the lowest mileage, the highest price, and the second-choice color. To quadruple the effect of the price on the decision factor, we can modify the equation as shown. cont’d

15 Example 1 – Solution This gives the results shown in Table 1.3. Although the numerical value of the decision factors changed, Car 6 still has the lowest decision factor. We decide to buy Car 6. cont’d Table 1.3

16 Mathematical Models Presented Numerically

17 Mathematical Models Presented Numerically Just as we may use equations to model a situation, we may also use a table of values. Whether in symbolic form (like a decision-factor equation) or numerical form (like a table of data), one of the purposes of a mathematical model is to make sense of the world around us.

18 Mathematical Models Presented Numerically Number of Registered Vehicles from 1980 to 2003

19 Example 2 – Analyzing a Mathematical Model in Numerical Form Describe the change in the number of registered vehicles in the United States. Then identify to whom this analysis may be important and why. Solution: The number of registered vehicles tends to be increasing. We can examine the amount of increase by subtracting one value from the next.

20 Example 2 – Solution The results are shown in Table 1.5. cont’d Table 1.5

21 Example 2 – Solution By examining these differences, we can describe how the number of registered vehicles is changing. From 1980 to 1990, the number increased by more than 33 million vehicles (about 3 million vehicles per year). From 1990 to 1995, the number increased by nearly 13 million vehicles (about 2.16 million vehicles per year). In the next 5-year interval (1995 to 2000), there was an increase of about 20 million vehicles (3.33 million vehicles per year). cont’d

22 Example 2 – Solution Although most drivers may not care about these statistics, many government agencies do. For example, state motor vehicle divisions that recognize these trends may be able to better forecast tax revenue from licensing fees. State and county officials may use these data to help shape plans for upgrades to transportation infrastructure such as roads, bridges, and highways. cont’d

23 Mathematical Models Presented Graphically

24 Mathematical Models Presented Graphically Newspapers and magazines present information in graphical form every day. Let’s use the context of median home prices in the metropolitan Phoenix area to investigate a mathematical model presented graphically. The metropolitan Phoenix area was one of the fastest growing areas in the United States in the early 2000s. As a result, home prices skyrocketed in 2005.

25 Mathematical Models Presented Graphically The graph in Figure 1.1 shows that the median price of homes in the Phoenix area increased from \$155,800 in the fourth quarter of 2003 to \$260,190 in the first quarter of 2006. Figure 1.1 Median Price of Homes—Phoenix Metro Area

26 Mathematical Models Presented Graphically In the graph, the fourth quarter of 2003 (October–December) is represented by t = 0. The first quarter of 2004 (January–March) is t = 1, the second quarter of 2004 (April–June) is t = 2, and so on. The vertical axis of the graph shows the median price in dollars.

27 Example 4 – Analyzing a Mathematical Model Presented Graphically Describe the trend seen in the graph (Figure 1.1) of the median home price in the metropolitan Phoenix area. Figure 1.1 Median Price of Homes—Phoenix Metro Area

28 Example 4 – Solution The median home prices increased from the fourth quarter of 2003 (t = 0) until the third quarter of 2005 (t = 7). Starting in quarter 7 (third quarter 2005), median home prices stabilized at approximately \$260,000. Median home prices increased very quickly between quarter 4 (fourth quarter 2004) and quarter 7 (third quarter 2005).