Finding a Linear Model Section 2.4. Lehmann, Intermediate Algebra, 4ed Section 2.4 A company’s profit was $10 million in 2005 and has increased by $3

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


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


Example Continued

Solution

Example Continued


• 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


Solution

Example Continued

Solution

3 10p t


• Verify the ordered pair (0, 10)• Verify that as the input increases by 1, the output

increases by 3

Slide 5


Graphing Calculator


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


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


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


Example Continued

Solution

Solution

0.04m

Example Continued


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


Example Continued

Solution

Solution

0.04 12G t

Example Continued

0.04 12G t


• 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


Solution Continued


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


• 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


Solution

Example Continued


• 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


Solution0.17 2.15s t y mx b

Example Continue


• Rates of change are shown in the table – all are close to 0.17

Slide 14


Solution


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


Solution

Example Continue

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Finding a Linear Model Section 2.4. Lehmann, Intermediate Algebra, 4ed Section 2.4 A company’s profit was $10 million in 2005 and has increased by $3