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

College Algebra Sixth Edition James Stewart  Lothar Redlin  Saleem Watson. Prerequisites. P. Modeling the Real World With Algebra. P.1. Introduction. In algebra, we use letters to stand for numbers. This allows us to describe patterns that we see in the real world. A Model for Pay.

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

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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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