Download PC-1.1a

Mathematical
Models 1.1a
Includes…geometric formulas,
regression analysis, solving
equations in 1 variable
Definitions
A mathematical model is a mathematical
structure that approximates phenomena for the
purpose of studying or predicting their behavior
One type of mathematical model: numerical
model, where numbers (or data) are analyzed to
gain insights into phenomena
Table 1.1 The Minimum Hourly Wage
Year
Min. Hourly Purchasing Power
Wage
in 2001 Dollars
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
0.30
0.30
0.75
0.75
1.00
1.25
1.60
2.10
3.10
3.35
3.35
4.25
5.15
3.68
2.88
5.43
4.79
5.82
6.84
7.23
6.88
6.80
5.48
4.57
4.94
5.34
1. How much did
Proctor get paid at
his first job?
2. What is the
national minimum
wage (as of
summer 2009)?
$7.25
Table 1.1 The Minimum Hourly Wage
Year
Min. Hourly Purchasing Power
Wage
in 2001 Dollars
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
0.30
0.30
0.75
0.75
1.00
1.25
1.60
2.10
3.10
3.35
3.35
4.25
5.15
3.68
2.88
5.43
4.79
5.82
6.84
7.23
6.88
6.80
5.48
4.57
4.94
5.34
3. In what five-year
period did the actual
minimum wage
increase the most?
Between 1975 and
1980, it increased
by $1.00
Table 1.1 The Minimum Hourly Wage
Year
Min. Hourly Purchasing Power
Wage
in 2001 Dollars
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
0.30
0.30
0.75
0.75
1.00
1.25
1.60
2.10
3.10
3.35
3.35
4.25
5.15
3.68
2.88
5.43
4.79
5.82
6.84
7.23
6.88
6.80
5.48
4.57
4.94
5.34
4. In what year did a
minimum-wage worker
have the greatest
purchasing power?
In 1970
5. What was the longest
period during which the
minimum wage did not
increase?
From 1940-1945,
1950-1955, and
1985-1990
Table 1.1 The Minimum Hourly Wage
Year
Min. Hourly Purchasing Power
Wage
in 2001 Dollars
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
0.30
0.30
0.75
0.75
1.00
1.25
1.60
2.10
3.10
3.35
3.35
4.25
5.15
3.68
2.88
5.43
4.79
5.82
6.84
7.23
6.88
6.80
5.48
4.57
4.94
5.34
6. A worker making min.
wage in 1980 was earning
nearly twice as much as a
worker making min. wage
in 1970  so why was there
pressure to once again
raise the min. wage?
Purchasing power
actually dropped by
$0.43 during that
period (inflation)
Table 1.1 The Minimum Hourly Wage
Year
Min. Hourly Purchasing Power
Wage
in 2001 Dollars
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
0.30
0.30
0.75
0.75
1.00
1.25
1.60
2.10
3.10
3.35
3.35
4.25
5.15
3.68
2.88
5.43
4.79
5.82
6.84
7.23
6.88
6.80
5.48
4.57
4.94
5.34
7. How many of you earn
the minimum hourly wage?
do you think that it is set
at a fair level?
Definitions
Another type of mathematical model:
Algebraic Model – uses formulas to relate
variable quantities associated with the
phenomena being studied
(Benefit: can generate numerical values of unknown
quantities 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 pizzas
are of the same thickness, which option gives the most pizza
for the money?
Calculate Areas:
18 24  432in
2
2
Circular pizza =  12   452.389in
Rectangular pizza =
2
The round pizza is larger, and is therefore the better deal
Guided Practice
At 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 is
the better value?
Calculate areas per dollar cost:
Small Pizza
A    5   25 in
2
Large Pizza
2
25 in
 19.635in 2 $
$4.00
2
A    7   49 in
2
2
49 in
2
 17.123in $
$8.99
2
Small: 19.635 in 2 /$, Large: 17.123 in 2/$
The small pizza is the better value!!!
Definition:
Another type of mathematical model:
Graphical Model – visual representation of a
numerical or algebraic model that gives insight
into the relationships between variable quantities
Regression Analysis:
The process of analyzing data by creating a scatter
plot, critiquing the data’s appearance (linear,
parabolic, cubic, etc.), choosing the appropriate
model, finding the line of best fit, making predictions
about the data….Handout!
A Good Example:
Galileo gathered data on a ball rolling down an inclined plane:
Elapsed Time
(seconds)
0
Distance
Traveled (in)
1
0 .75
2
3
3
4
5
6
6.75 12 18.75 27
7
8
36.75
48
1. Create a scatter plot of these data
2. Derive an algebraic model to fit these data
d = 0.75t 2
3. 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
Distance
Traveled (in)
1
0 .75
2
3
3
4
5
6
6.75 12 18.75 27
7
8
36.75
48
4. How far will the ball have traveled after 15 seconds?
d = 168.75 in
5. How long will it take the ball to travel 62 inches?
t = 9.092 sec
More Practice Problems…
Find all real numbers x for which 6x 3 = 11x 2 + 10x
6 x  11x  10 x  0
3
2
x  6 x  11x  10   0
2
x  2 x  5 3x  2  0
5
x = 0 or x =
2
2
or x = –
3
We just used the Zero Factor Property:
A product of real numbers is zero if and only if at least one
of the factors in the product is zero.
Terminology:
If a is a real number that solves the equation f(x) = 0, then
these 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
Solve the equation algebraically and graphically.
 x  11
2
 121
x  22, 0
Guided Practice
Solve the equation algebraically and graphically.
3
x  7x   0
4
2
7
x   13  0.106, 7.106
2
Guided Practice..
Solve the equation algebraically and graphically.
z  2 z  1  10
z  2, 2.5
Guided Practice
Solve the equation algebraically and graphically.
x  x 1
Use the quadratic formula:
x  1 x
 x
2
 1  x 
x  1 2x  x
2
2
0  x  3x  1
2
3
5
x 
2 2
Check for extraneous solutions!!!
3
5
x 
 0.382
2 2
Whiteboard if time…
#24 on p.77-78
(a) Scatterplot window:
1,14 by 400,700
(b) Graph in same window:
P  1.13x  3.1x  443
2
 3.1x  443  900
x  21.529,18.765  In 2005
(c) Solve: 1.13 x
2
(d) No, the algebraic model will probably not be valid in
future years…why not?
Homework: p. 76-78 1-17 all, 29-37 odd