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Modeling the Real World with Algebra: Introduction to Algebra Models

Learn how to use algebraic models to describe real-world patterns and solve practical problems. This textbook introduces the basics of modeling with algebra and provides examples to enhance understanding.

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Modeling the Real World with Algebra: Introduction to Algebra Models

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  1. College Algebra Sixth Edition James StewartLothar RedlinSaleem Watson

  2. Prerequisites P

  3. Modeling the Real WorldWith Algebra P.1

  4. Introduction • In algebra, we use letters to stand for numbers. • This allows us to describe patterns that we see in the real world.

  5. A Model for Pay • For example, if we let N stand for the number of hours that you work and W stand for your hourly wage, then the formulaP = NWgives your pay P.

  6. A Model for Pay • The formula P = NW is a description or model for pay. • We can also call this formulaan algebra model.

  7. A Model for Pay • We summarize the situation as follows:

  8. A Model for Pay • The model P = NW gives a pattern for finding • The pay for any worker, • With any hourly wage, • Working any number of hours.

  9. A Model for Pay • That’s the power of algebra: • By using letters to stand for number, we can write a single formula that describes many different situations.

  10. A Model for Pay • We can now use the model P = NW to answer questions such as • “I make $10 an hour, and I worked35 hours; how much do I get paid?” or • “I make $8 an hour; how many hoursdo I need to work to get paid $1000?”

  11. Models and Modeling • In general, a model is a mathematical representation (such as a formula) of a real-world situation. • Modeling is the process of making mathematical models.

  12. Models and Modeling • Once a model has been made, it can beused to answer questions about the thing being modeled.

  13. Section Overview • The examples we study in this section are simple. • But, the methods are far reaching. • This will become more apparent as we explorethe applications of algebra in subsequent Focus onModeling sections that follow each chapter.

  14. Using Algebra Models

  15. Algebra Models • We begin our study of modeling by using models that are given to us. • In the next subsection we learn how to makeour own models.

  16. E.g. 1—Using a Model for Pay • Aaron makes $9 an hour at his part-time job. • Use the model P = NW:

  17. E.g. 1—Using a Model for Pay • Aaron worked 35 hours last week. • How much did he get paid? • Aaron wants to earn enough money to buy a calculus text that costs $126. • How many hours does he need to work to earn this amount?

  18. Example (a) E.g. 1—Using a Model for Pay • We know that N = 35 h and W = $9. • To find P, we substitute these values into the formula. P = NW= 35 × 9 = 315 • So Aaron was paid $315.

  19. Example (b) E.g. 1—Using a Model for Pay Aaron's hourly wage is W = $9, and the amount of pay he needs to buy the book is P = $216. • To find N, we substitute these values into the model.P = NW126 = 9N 126/9 = N N = 14 • So Aaron must work 14 hours to buy this book.

  20. E.g. 2—Using an Elevation-Temperature Model • A mountain climber uses the model • T = 20 – 10h • to estimate the temperature T (in °C) at elevation h (in kilometers, km).

  21. E.g. 2—Using an Elevation-Temperature Model • Make a table that gives the temperature for each 1-km change in elevation. • Go from elevation 0 km to elevation 5 km. • How does temperature change as elevation increases? • If the temperature is 5°C, what is the elevation?

  22. Example (a) E.g. 2—Elevation-Temperature • Let’s use the model to find the temperature at elevation h = 3 km. • T = 20 – 10h • = 20 – 10(3) • = –10 • So at an elevation of 3 km the temperature is –10°C.

  23. Example (a) E.g. 2—Elevation-Temperature • The other entries in the following table are calculated similarly. • We see that temperature decreases as elevation increases.

  24. Example (b) E.g. 2—Elevation-Temperature • We substitute T = 5°C in the model and solve for h: • T = 20 – 10h • 5 = 20 – 10h • –15 = –10h • –15/–10 = h • 1.5 = h • The elevation is 1.5 km.

  25. Making Algebra Models

  26. Making Algebra Models • In the next example, we explore the process of making an algebra model for a real-life situation.

  27. E.g. 3—Making a Model for Gas Mileage • The gas mileage of a car is the number of miles it can travel on one gallon of gas. • Find a formula that models gas mileage in terms of the number of miles driven and the number of gallons of gasoline used. • Henry’s car used 10.5 gallons to drive 230 miles. Find its gas mileage.

  28. E.g. 3—Making a Model for Gas Mileage • Let’s try a simple case. • If a car uses 2 gallons to drive 100 miles,we easily see that gas mileage = 100/2 = 50 mi/gal • So gas mileage is the number of milesdriven divided by the number of gallonsused.

  29. Example (a) E.g. 3—A Model for Gas Mileage • To find the formula we want, we need to assign symbols to the quantities involved:

  30. Example (a) E.g. 3—A Model for Gas Mileage • We can express the model as follows: • gas mileage = number of miles driven • number of gallons used • M = N/G

  31. Example (b) E.g. 3—A Model for Gas Mileage • To get the gas mileage, we substituteN = 230 and G = 10.5 in the formula: • M = N/G • = 230/10.5 • ≈ 21.9 • The gas mileage for Henry’s car is about21.9 mi/gal.

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