[email protected] MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

[email protected] • MTH15_Lec-04_sec_1-4_Functional_Models_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§1.4 MathModels



Review §

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

1.3



§1.4 Learning Goals

Study general modeling procedure

Explore a variety of applied models

Investigate market equilibrium and break-even analysis in economics



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



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

Symbol Ra

Atomic Number 88

Atomic Mass 226.0254

Electron Configuration 2.8.18.32.18.8.2 [Rn].7s2

Valence Number 2

Oxidation Numbers +2

Melting Point 973°K, 700°C, 1292°F

Boiling Point 1809°K, 1536°C, 2797°F

Family 2

Series 7

Element Classification Alkali Earth Metal

Density 5.5g/cc @ 300K

Crystal Structure body-centred cubic

State of Matter Solid

Date/Place of Discovery 1898, France

Person Who Discovered Pierre and Marie Curie

Ra Elemental Facts:





Marketing of Products A & B

Profit Fcn given x% of Marketing Budget Spent on product A:

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

10072

7230

300

for

for

for

25.080

5.026

7.020

x

x

x

x

x

x

xP



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

x (%) y = P(x)0 2030 4130 4172 6272 62100 55



Th

e Plo

t (By M

AT

LA

B)

0 20 40 60 80 1000

10

20

30

40

50

60

70

x

P(x

)MTH15 • Bruce Mayer, PE • P1.4-22

M15P010422Marketing1306.mm



Pro

fit for x =

50%

0 20 40 60 80 1000

10

20

30

40

50

60

70

x

P(x

)MTH15 • Bruce Mayer, PE • P1.4-22

M15P010422Marketing1306.mm

50

51

512526505.02650 P







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



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 (curve-

fits) 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



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 Find

• Total Revenue, R(x)• Total Cost, C(x)• BreakEven Volume

xxR

card

75.2$

kxxC 12$card

35.0$

kxCRxP 12$card

35.0$

card

75.2$



R &

C P

lot

0 1000 2000 3000 4000 5000 6000 7000 80000

2

4

6

8

10

12

14

16

18

20

22

x (cards)

R&

C (

$k)

MTH15 • Bruce Mayer, PE • P1.4-38

Revenue

Cost

M15P1438GreetingCardProf itPlot1306.mM15P1438GreetingCardProf itPlot1306.mM15P1438GreetingCardProf itPlot1306.m

Break Even



P &

L Z

on

es

0 1000 2000 3000 4000 5000 6000 7000 80000

2

4

6

8

10

12

14

16

18

20

22

x (cards)

R&

C (

$k)


M15P1438GreetingCardProf itPlot1306.m

LOSSZone

ProfitZone



MA

TL

AB

cod

e

% 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 Functionxmin = 0; xmax = 8000; % in Cardsymin = 0; ymax = 22000 % in $;x = linspace(xmin,xmax,500);S = 2.75 % $k/cardCv = 0.35 % $/cardCf = 12000 % $R = S*x; C = Cv*x + Cf;P = R - C; %% Use fzero to find Crossing PointZfcn = @(u) S*u - (Cv*u + Cf)% Check Zereos by Ploty3 = Zfcn(x);plot(x, y3,[0,xmax], [0,0], 'LineWidth', 3),grid, title(['\fontsize{16}ZERO Plot',])display('Showing fcn ZERO Plot; hit ANY KEY to Continue')pause%% Find ZerosxE = fzero(Zfcn,[4000 6000])PE = S*xE - (Cv*xE + Cf) plot(x,R/1000, x,C/1000, 'k','LineWidth', 2), axis([0 xmax ymin ymax/1000]),... grid, xlabel('\fontsize{14}x (cards)'), ylabel('\fontsize{14}R&C ($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 graphshold onpause%xn = linspace(xmin, xmax, 100);fill([xn,fliplr(xn)],[S*xn/1000, fliplr(Cv*xn + Cf)/1000],'m')











P1.4-60 Build a Box

Given 18” Square of CardBoard, then Construct Largest Volume Box

18”

x

x

http://www.google.com/imgres?imgurl&imgrefurl=http://stickinsect.wordpress.com/2011/01/22/making-a-cardboard-box/&h=0&w=0&sz=1&tbnid=jMQOV3kpx1I96M&tbnh=181&tbnw=278&prev=/search?q=build+a+box&tbm=isch&tbo=u&zoom=1&q=build%20a%20box&docid=eYcqZDLYtG7UVM&hl=en&ei=op_MUd2LAuKWiALxxYGgCg&ved=0CAEQsCU





Larg

est Bo

x

0 1 2 3 4 5 6 7 8 90

50

100

150

200

250

300

350

400

450

Box Height, x (inches)

Bo

x V

olu

me

, V (

inch

es3 )


MTH15P1460BoxConstructionVolume1306.m

432

xxxxV 32436 23



MA

TL

AB

& M

uP

AD

% Bruce Mayer, PE% MTH-15 • 23Jun13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m% ref:%clear; clc%% The Limitsxmin = 0; xmax = 9; ymin = 0; ymax = 450;% The FUNCTIONx = 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 PointImax = find(y>=max(y)); Vmax = max(y), Xmax = x(Imax)%% the Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 • P1.4-60',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15P1460BoxConstructionVolume1306.m','FontSize',7)

q := x*(18-x)^2

Simplify(q)

expand(q)



Surf Area Prob

Find the SurfaceArea for this Solid

Find By SUBTRACTION

=

+

NEW Exposed Surface



Surf Area Prob cont.1

The Box Surf. Area

B = 4-Sides + [Top & Bot]

B = 4•xh + 2•x2

The Cylinder Area

C = [Top & Bot] − Sides

C = 2•πr2 − π•(2r)•h



Surf Area P cont.2

Then the NET Surface Area, S, by

S = B – C

= [4xh + 2x2] – [2•πr2 – π•(2r)•h]

= 2x2– 2πr2 + 2πrh + 4xh

= +S B C



All Done for Today

FluidMechanics

Math Model



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=iyS8Qis0aGh1OM&tbnid=Vr8-lHLIzG7XjM:&ved=0CAgQjRwwAA&url=http://exploration.grc.nasa.gov/education/rocket/nseqs.html&ei=4LrMUYmeH4jviQKdyoHwBw&psig=AFQjCNGOAJqvusgeYX0kpnddIz01dp58EQ&ust=1372458080550751

[email protected] MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

Documents

pe chabot college mathematics1

modelsall math models

view math models

physicallaw models

cfd physical models

variety of applied models

okhumanjudgment models

n s j o h n s o n c