Chabot Mathematics
§1.4 Math
Models
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Review §
1.3
 Any QUESTIONS About
• §1.3 → Lines & LinearFunctions
 Any QUESTIONS AboutHomeWork
• §1.3 → HW-03
 h ≡ Si PreFix for 100X; e.g.:
• $100 = $h
• 100 Units = hU
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
§1.4 Learning Goals
 Study general modeling procedure
 Explore a variety of applied models
 Investigate market equilibrium and
break-even analysis in economics
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Functional Math Modelling
 Mathematical modeling is the process of
translating statements in WORDS &
DIAGRAMS into equivalent statements
in mathematics.
• This Typically an
ITERATIVE Process;
the model is
continuously
adjusted to produce
Real-World Results
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
P1.4-10: Radium Decay Rate
 A Sample of Radium
(Ra) decays at a rate,
RRa, that is
ProPortional to the
amount of Radium, mRa,
Remaining
 Express the Rate of
Decay, RRa, as a
function of the
ReMaining Amount, mRa
Chabot College Mathematics
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Ra Elemental Facts:
Symbol
Ra
Atomic Number
88
Atomic Mass
Electron Configuration
Valence Number
226.0254
2.8.18.32.18.8.2
[Rn].7s2
2
Oxidation Numbers +2
Melting Point
Boiling Point
973°K, 700°C, 1292°F
1809°K, 1536°C, 2797°F
Family
2
Series
7
Element Classification
Density
Crystal Structure
State of Matter
Date/Place of Discovery
Person Who Discovered
Alkali Earth Metal
5.5g/cc @ 300K
body-centred cubic
Solid
1898, France
Pierre and Marie Curie
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Marketing of Products A & B
 Profit Fcn given x% of Marketing Budget
Spent on product A:
Px 
20  0.7 x
 26  0.5 x
80  0.25 x
for
for
for
0  x  30
30  x  72
72  x  100
a. Sketch Graph
b. Find P(50) for 50-50 marketing expense
c. Find P(y) where y is the % of Markeing
Budget expended on Product B
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Marketing of Products A & B
 Make T-Table to
Sketch Graph
 Note that only END
POINTS of lines are
needed to plot
piece-wise Linear
Function
Chabot College Mathematics
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x (%) y = P(x)
0
20
30
41
30
41
72
62
72
62
100
55
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
The Plot (By MATLAB)
MTH15 • Bruce Mayer, PE • P1.4-22
70
60
P(x)
50
40
30
20
10
0
M15P010422Marketing1306.mm
0
Chabot College Mathematics
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20
40
60
x
80
100
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
MTH15 • Bruce Mayer, PE • P1.4-22
70
P50  26  0.550  26  25  51
Profit for x = 50%
60
P(x)
51
50
40
30
20
10
0
M15P010422Marketing1306.mm
0
Chabot College Mathematics
10
20
40
50
x
60
80
100
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Caveat Exemplars (Beware Models)
 Q) From WHERE do these Math
Models Come?
 A) From PEOPLE; Including Me and
YOU!
 View Math Models with Considerable
SKEPTISISM!
• Physical-Law Models are the Best
• Statistical Models (curve fits) are OK
• Human-Judgment Models are WORST
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Caveat Exemplars (Beware Models)
 ALL Math Models MUST be verified
against RealWorld RESULTS; e.g.:
• CFD (Physical) Models Checked by Wind
Tunnel Testing at NASA-Ames
• Biology species-population models (curvefits) tested against field observations
• Stock-Market Models are discarded often
 LEAST Reliable models are those that
depend on HUMAN BEHAVIOR (e.g.
Econ Models) that can Change Rapidly
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
P1.4-38 Greeting Card BreakEven
 Make & Sell Greeting Cards
• Sell Price, S = $2.75/card
• Fixed Costs, Cf = $12k
• Variable Costs, Cv = $0.35/Card
 Let x ≡ Number of Cards Rx    $2.75   x
 card 
 Find
• Total Revenue, R(x)
• Total Cost, C(x)
 $0.35 
C x   
 x  $12k

 card 
 $2.75 $0.35 
Px   R  C  

 x  $12k

card 
 card
• BreakEven Volume
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
MTH15 • Bruce Mayer, PE • P1.4-38
22
20
Break
Even
18
16
R & C Plot
R&C ($k)
14
12
10
8
6
4
Revenue
Cost
2
0
M15P1438GreetingCardProf itPlot1306.m
0
1000
2000
3000
4000
5000
6000
7000
8000
x (cards)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
MTH15 • Bruce Mayer, PE • P1.4-38
22
20
18
Profit
Zone
16
P & L Zones
R&C ($k)
14
12
10
LOSS
Zone
8
6
4
2
0
M15P1438GreetingCardProf itPlot1306.m
0
1000
2000
3000
4000
5000
6000
7000
8000
x (cards)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
%
% Use fzero to find Crossing Point
Zfcn = @(u) S*u - (Cv*u + Cf)
% Check Zereos by Plot
y3 = Zfcn(x);
plot(x, y3,[0,xmax], [0,0], 'LineWidth', 3),grid, title(['\fontsize{16}ZERO Plot',])
display('Showing fcn ZERO Plot; hit ANY KEY to Continue')
pause
%
% Find Zeros
xE = fzero(Zfcn,[4000 6000])
PE = S*xE - (Cv*xE + Cf)
plot(x,R/1000, x,C/1000, 'k','LineWidth', 2), axis([0 xmax ymin ymax/1000]),...
grid, xlabel('\fontsize{14}x (cards)'), ylabel('\fontsize{14}R&C ($k)'),...
title(['\fontsize{16}MTH15 • Bruce Mayer, PE • P1.4-38',]),...
annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none',
'String', 'M15P1438GreetingCardProfitPlot1306.m','FontSize',7)
display('Showing 2Fcn Plot; hit ANY KEY to Continue')
% "hold" = Retain current graph when adding new graphs
hold on
pause
%
xn = linspace(xmin, xmax, 100);
fill([xn,fliplr(xn)],[S*xn/1000, fliplr(Cv*xn + Cf)/1000],'m')
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
MATLAB code
% Bruce Mayer, PE
% MTH-15 • 27Jun13
% M15_P14_38_Greeting_Card_Profit_Plot_1306.m
% Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E.
% Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN
% 978-0-13-199110-1, Pearson Higher Ed, 2011, pp294-295
%
clc; clear
% The Function
xmin = 0; xmax = 8000; % in Cards
ymin = 0; ymax = 22000 % in $;
x = linspace(xmin,xmax,500);
S = 2.75 % $k/card
Cv = 0.35 % $/card
Cf = 12000 % $
R = S*x; C = Cv*x + Cf;
P = R - C;
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
P1.4-60  Build a Box
 Given 18” Square of CardBoard, then
Construct Largest Volume Box
18”
x
x
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
MTH15 • Bruce Mayer, PE • P1.4-60
450
432
V x   x 3  36 x 2  324 x
350
300
Largest Box
Box Volume, V (inches3)
400
250
200
150
100
50
0
MTH15P1460BoxConstructionVolume1306.m
0
1
Chabot College Mathematics
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2
3
4
5
6
Box Height, x (inches)
7
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
9
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
MATLAB & MuPAD
% Bruce Mayer, PE
q := x*(18-x)^2
% MTH-15 • 23Jun13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
% ref:
Simplify(q)
%
clear; clc
%
expand(q)
% The Limits
xmin = 0; xmax = 9;
ymin = 0; ymax = 450;
% The FUNCTION
x = linspace(xmin,xmax,500); y = x.*(18-2*x).^2;
%
% The ZERO Lines +> Do not need this time
% * zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin
ymax];
%
% FIND the Max Point
Imax = find(y>=max(y)); Vmax = max(y), Xmax = x(Imax)
%
% the Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y, Xmax,Vmax, 'p' , 'LineWidth', 3),axis([xmin xmax ymin
ymax]),...
grid, xlabel('\fontsize{14}Box Height, x (inches)'),
ylabel('\fontsize{14}Box Volume, V (inches^3)'),...
title(['\fontsize{16}MTH15 • Bruce Mayer, PE • P1.460',]),...
annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on',
'EdgeColor', 'none', 'String',
'MTH15P1460BoxConstructionVolume1306.m','FontSize',7)
Surf Area Prob
 Find the Surface
Area for
this Solid
 Find By SUBTRACTION
=
+
NEW Exposed Surface
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Surf Area Prob cont.1
 The Box Surf. Area
 The Cylinder Area
 B = 4-Sides +
[Top & Bot]
 B = 4•xh + 2•x2
 C = [Top & Bot] −
Sides
 C = 2•πr2 − π•(2r)•h
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Surf Area P cont.2
 Then the NET Surface Area, S, by
S
=
B
+
 S=B–C
= [4xh + 2x2] – [2•πr2 – π•(2r)•h]
= 2x2– 2πr2 + 2πrh + 4xh
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
C
All Done for Today
Fluid
Mechanics
Math Model
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx