[email protected] mth15_lec-04_sec_1-4_functional_models_.pptx 1 bruce mayer, pe chabot...

[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



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[email protected] mth15_lec-04_sec_1-4_functional_models_.pptx 1 bruce mayer, pe chabot...

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