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Finding a Linear Model
Section 2.4
Lehmann, Intermediate Algebra, 4edSection 2.4
A company’s profit was $10 million in 2005 and has increased by $3 million per year. Let p be the profit (in millions of dollars) in the year that is t years since 2005.
1. Is there a linear relationship between t and p?
•Since the profit is a constant $3 million per year, the variables t and p are linearly related
Slide 2
Finding a ModelFinding an Equation of a Linear Model
Example
Solution
Lehmann, Intermediate Algebra, 4edSection 2.4
2. Find the p-intercept of a linear model
• Profit was $10 million in 2005• 2005 is 0 years since 2005• This gives the ordered pair (0, 10)• So, the p-intercept is (0, 10)
3. Find the slope of the linear model.
Slide 3
Finding a ModelFinding an Equation of a Linear Model
Example Continued
Solution
Example Continued
Lehmann, Intermediate Algebra, 4edSection 2.4
• Rate of change of profit per year is $3 million per year• So, the slope of the linear model is 3
4.Find an equation of the linear model.
• Since p-intercept is (0, 10) and slope is 3, the linear model is:
Slide 4
Finding a ModelFinding an Equation of a Linear Model
Solution
Example Continued
Solution
3 10p t
Lehmann, Intermediate Algebra, 4edSection 2.4
• Verify the ordered pair (0, 10)• Verify that as the input increases by 1, the output
increases by 3
Slide 5
Finding a ModelFinding an Equation of a Linear Model
Graphing Calculator
Lehmann, Intermediate Algebra, 4edSection 2.4
We perform a unit analysis of a model’s equations by determining the units of the expression on both sides of the equation. The simplified units of the expressions on both sides of the equation should be the same.
Slide 6
Definition: Unit AnalysisUnit Analysis of a Linear Model
Definition
Lehmann, Intermediate Algebra, 4edSection 2.4
A driver fills her car’s 12-gallon gasoline tank and drives as a constant speed. The car consumes 0.04 gallon per mile. Let G be the number of gallons of gasoline remaining in the tank after she has driven d miles since filling up.
1. Is there a linear relationship between d and G?
•Rate of change is a constant –0.04 gallons per minute, so d and G are linearly related
Slide 7
Finding a ModelUnit Analysis of a Linear Model
Example
Solution
Lehmann, Intermediate Algebra, 4edSection 2.4
2. Find the G-intercept of a linear model.
• Tank is full at 12 gallons: ordered pair (0, 12)
3. Find the slope of the linear model.
• Gasoline remaining in the tank with respect to distance traveled is:
Slide 8
Finding a ModelUnit Analysis of a Linear Model
Example Continued
Solution
Solution
0.04m
Example Continued
Lehmann, Intermediate Algebra, 4edSection 2.4
4. Find the equation of the linear model.
• Since p-intercept is (0, 12) and slope is –0.04, the linear model is:
5. Perform a unit analysis of the equation
• Here a unit analysis on the equation :
Slide 9
Finding a ModelUnit Analysis of a Linear Model
Example Continued
Solution
Solution
0.04 12G t
Example Continued
0.04 12G t
Lehmann, Intermediate Algebra, 4edSection 2.4
• We use the fact that = 1 to simplify the units of
the expression on the right-hand side of the equation:
• Units on both sides are gallons: Suggesting correct
Slide 10
Finding a ModelUnit Analysis of a Linear Model
Solution Continued
Lehmann, Intermediate Algebra, 4edSection 2.4
Yogurt sales (in billions of dollars) in the United States are shown in the table for various years.
Let s be yogurt sales (in billions of dollars) in the year that is t years since 2000.
A model of the situation is:
1. Use a graphing calculator to draw a scattergram and the model in the same viewing window. Check whether the line comes close to the data.
Slide 11
Analyzing a ModelTwo Variables That Are Approximately Linearly Related
Example
0.17 2.15s t
Lehmann, Intermediate Algebra, 4edSection 2.4
• Draw in the same screen using a graphing calculator• See Sections B.8 and B.10
2. What is the slope of the model? What does it mean in this situation?
Slide 12
Analyzing a ModelTwo Variables That Are Approximately Linearly Related
Solution
Example Continued
Lehmann, Intermediate Algebra, 4edSection 2.4
• which is of the form • Since m is the slope, the slope is 0.17• Sales increase by 0.17 billion dollars per year
3. Find the rates of change in sales from one year to the next. Compare the rates of change with the results in Problem 2.
Slide 13
Analyzing a ModelTwo Variables That Are Approximately Linearly Related
Solution0.17 2.15s t y mx b
Example Continue
Lehmann, Intermediate Algebra, 4edSection 2.4
• Rates of change are shown in the table – all are close to 0.17
Slide 14
Analyzing a ModelTwo Variables That Are Approximately Linearly Related
Solution
Lehmann, Intermediate Algebra, 4edSection 2.4
4. Predict the sales in 2010.
• Substitute the input of 10 for t:
If two quantities t and p are approximately linearly related, and if p depends on t, then the slope of a reasonable linear model is approximately equal to the average rate of change of p with respect to t.
Slide 15
Analyzing a ModelTwo Variables That Are Approximately Linearly Related
Solution
Example Continue
Property