MathematicaMathematical Models l Models 1.1a1.1aIncludes…geometric formulas, Includes…geometric formulas, regression analysis, solving regression analysis, solving equations in 1 variableequations in 1 variable

DefinitionsDefinitions

A A mathematical modelmathematical model is a mathematical is a mathematicalstructure that approximates phenomena for thestructure that approximates phenomena for thepurpose of studying or predicting their behaviorpurpose of studying or predicting their behavior

One type of mathematical model: One type of mathematical model: numericalnumericalmodelmodel, where numbers (or , where numbers (or datadata) are analyzed to) are analyzed togain insights into phenomenagain insights into phenomena

Table 1.1 The Minimum Hourly Wage

Year Min. Hourly Purchasing Power Wage in 2001 Dollars

1940 0.30 3.681945 0.30 2.881950 0.75 5.431955 0.75 4.791960 1.00 5.821965 1.25 6.841970 1.60 7.231975 2.10 6.881980 3.10 6.801985 3.35 5.481990 3.35 4.571995 4.25 4.942000 5.15 5.34

What is thenational minimum

wage (as ofsummer 2009)?

$7.25$7.25

Table 1.1 The Minimum Hourly Wage

Year Min. Hourly Purchasing Power Wage in 2001 Dollars

1940 0.30 3.681945 0.30 2.881950 0.75 5.431955 0.75 4.791960 1.00 5.821965 1.25 6.841970 1.60 7.231975 2.10 6.881980 3.10 6.801985 3.35 5.481990 3.35 4.571995 4.25 4.942000 5.15 5.34

In what five-yearperiod did the actual

minimum wageincrease the most?

Between 1975 andBetween 1975 and1980, it increased1980, it increased

by $1.00by $1.00

Table 1.1 The Minimum Hourly Wage

Year Min. Hourly Purchasing Power Wage in 2001 Dollars

1940 0.30 3.681945 0.30 2.881950 0.75 5.431955 0.75 4.791960 1.00 5.821965 1.25 6.841970 1.60 7.231975 2.10 6.881980 3.10 6.801985 3.35 5.481990 3.35 4.571995 4.25 4.942000 5.15 5.34

In what year did aminimum-wage worker

have the greatestpurchasing power?

In 1970In 1970

What was the longestperiod during which theminimum wage did not

increase?

From 1940-1945,From 1940-1945,1950-1955, and1950-1955, and

1985-19901985-1990

Table 1.1 The Minimum Hourly Wage

Year Min. Hourly Purchasing Power Wage in 2001 Dollars

1940 0.30 3.681945 0.30 2.881950 0.75 5.431955 0.75 4.791960 1.00 5.821965 1.25 6.841970 1.60 7.231975 2.10 6.881980 3.10 6.801985 3.35 5.481990 3.35 4.571995 4.25 4.942000 5.15 5.34

A worker making min.wage in 1980 was earningnearly twice as much as aworker making min. wage

in 1970 so why was therepressure to once againraise the min. wage?

Purchasing powerPurchasing poweractually dropped byactually dropped by$0.43 during that$0.43 during thatperiod (inflation)period (inflation)

Table 1.1 The Minimum Hourly Wage

Year Min. Hourly Purchasing Power Wage in 2001 Dollars

1940 0.30 3.681945 0.30 2.881950 0.75 5.431955 0.75 4.791960 1.00 5.821965 1.25 6.841970 1.60 7.231975 2.10 6.881980 3.10 6.801985 3.35 5.481990 3.35 4.571995 4.25 4.942000 5.15 5.34

How many of you earnthe minimum hourly wage?do you think that it is set

at a fair level?

DefinitionsDefinitions

Another type of mathematical model:Another type of mathematical model:Algebraic ModelAlgebraic Model – uses – uses formulasformulas to relate to relatevariable quantities associated with thevariable quantities associated with thephenomena being studiedphenomena being studied

(Benefit: can generate numerical values of unknownquantities using known quantities)

Guided Practice:A restaurant sells a rectangular 18” by 24” pizza for the same

price as its large round pizza (24” diameter). If both pizzasare of the same thickness, which option gives the most pizzafor the money?

The round pizza is larger, and is therefore the better dealThe round pizza is larger, and is therefore the better deal

Calculate Areas:

Rectangular pizza = 18 24 2432inCircular pizza = 212 2452.389in

Guided PracticeAt Dominos, a small (10” diameter) cheese pizza costs $4.00,while a large (14” diameter) cheese pizza costs $8.99.Assuming that both pizzas are the same thickness, which isthe better value?

The small pizza is the better value!!!The small pizza is the better value!!!

Small: 19.635 in /$, Large: 17.123 in /$Small: 19.635 in /$, Large: 17.123 in /$2222

Calculate areas per dollar cost:

Small Pizza

2 25 25 inA 225 in

$4.00

219.635in $

Large Pizza

2 27 49 inA 249 in

$8.99

217.123in $

Definition: Another type of mathematical model:Another type of mathematical model:Graphical ModelGraphical Model – visual representation of a– visual representation of anumerical or algebraic model that gives insightnumerical or algebraic model that gives insightinto the relationships between variable quantitiesinto the relationships between variable quantities

Regression Analysis: The process of analyzing data by creating a scatterThe process of analyzing data by creating a scatterplot, critiquing the data’s appearance (linear, plot, critiquing the data’s appearance (linear, parabolic, cubic, etc.), choosing the appropriate parabolic, cubic, etc.), choosing the appropriate model, finding the line of best fit, making predictionsmodel, finding the line of best fit, making predictionsabout the data….Handout!about the data….Handout!

A Good Example:

Galileo gathered data on a ball rolling down an inclined plane:

Elapsed Time (seconds) 0 1 2 3 4 5 6 7 8

Distance Traveled (in)

0 .75 3 6.75 12 18.75 27 36.75 48

1. Create a scatter plot of these data1. Create a scatter plot of these data

2. Derive an algebraic model to fit these data2. Derive an algebraic model to fit these data

d = 0.75td = 0.75t22

3. Graph this function on top of your scatter plot3. Graph this function on top of your scatter plot

A Good Example:

Galileo gathered data on a ball rolling down an inclined plane:

Elapsed Time (seconds) 0 1 2 3 4 5 6 7 8

Distance Traveled (in)

0 .75 3 6.75 12 18.75 27 36.75 48

4. How far will the ball have traveled after 15 seconds?4. How far will the ball have traveled after 15 seconds?

5. How long will it take the ball to travel 62 inches?5. How long will it take the ball to travel 62 inches?

d = 168.75 ind = 168.75 in

t = 9.092 sect = 9.092 sec

More Practice Problems…

Find all real numbers x for which 6x = 11x + 10x23

x = 0 or x = or x = –5

2

2

3

We just used the Zero Factor PropertyZero Factor Property:A product of real numbers is zero if and only if at least oneof the factors in the product is zero.

3 26 11 10 0x x x

26 11 10 0x x x

2 5 3 2 0x x x

Terminology:

If a is a real number that solves the equation f(x) = 0, thenthese three statements are equivalent:

1. The number a is a root (or solution) of the equation f(x) = 0.

2. The number a is a zero of y = f(x).

3. The number a is an x-intercept of the graph of y = f(x). (sometimes the point (a, 0) is referred to as an x-intercept)

Guided Practice

211 121x Solve the equation algebraically and graphically.

22,0x

Guided Practice

2 37 0

4x x

Solve the equation algebraically and graphically.

713 0.106,7.106

2x

Guided Practice..

2 1 10z z Solve the equation algebraically and graphically.

2,2.5z

Guided PracticeSolve the equation algebraically and graphically.

Check for extraneous solutions!!!Check for extraneous solutions!!!

1x x

1x x

2

21x x 21 2x x x

20 3 1x x

Use the quadratic formula:

3 5

2 2x

3 50.382

2 2x

Whiteboard if time…#24 on p.77-78

(a) Scatterplot window: 1,14 by 400,700(b) Graph in same window: 21.13 3.1 443P x x (c) Solve: 21.13 3.1 443 900x x

21.529,18.765x In 2005

(d) No, the algebraic model will probably not be valid infuture years…why not?

Homework: p. 76-78 1-17 all, 29-37 odd