corralling ideas on optimization lalu simcik, phd cabrillo college amatyc 2009 lsimcik
Post on 25-Dec-2015
224 Views
Preview:
TRANSCRIPT
Corralling ideas on optimization
Lalu Simcik, PhD
Cabrillo College
AMATYC 2009
www.cabrillo.edu/~lsimcik
The corral problem
• Rectangular corral with constrained length of fence (say 1000 feet)
• Perimeter equation
• Area Equation transformed to area function with variable substitution
xwwx 500100022
xxxA
xxxA
xwA
500
)500(2
The corral problem
• Vertex of a parabola– Midpoint of the quadratic
formula roots – completing the square– Uniqueness
• For one animal
• Leads to proof that the ideal rectangle is a square (single corral case)
500
12
b
aa
bxv
feetwx 250
Area function Parabola
500
12
b
aa
bxv
feetwx 500
xxxA 5002
feetxw
feetx
250500
250
Got more animals?
Variations
• Two animals• Three animals
• Two animals by the river• Three animals by the
river
• Is there a pattern in all these examples?
More variations
What is the pattern in all of these examples?
Algebra II
• Simplified, step by step presentation
• Offer A(x) to use even if student is blocked
• Prioritize use of vertex
• Avoid using ‘y’
7. Llama Corral by the River: A farmer has 1000 ft of fence to make a rectangular Llama corral by a river (see picture). w x x River a) Create an equation that adds up fence lengths to equal the 1000 ft. (3 pts) b) Solve your equation for ‘w’ in part a) (2 pts) c) Use part b) to show that the Area Function (which is a function of x) is: (7 pts) xxxA 10002)( 2 d) Find the length in feet for x that will maximize the area of the corral. After you find this x, use it to find the length of w. (Hint: vertex!)
(5 pts)
Presentation in Precalculus
• More autonomous style
• Double Jeopardy
• More animals
3. With 1000 ft of fence, find the dimensions for a rectangular corral that maximize the area for 2 animals by the river. (see drawing on board) (8 pts) ↓↓↓ iii) Find the best x and w (7 pts) ↓↓↓
i) Perimeter equation:
ii) Find )(xA
)(xA = _________________
Presentation in Calculus I
• n-Animals
• Using related rates in lieu of variable substitution
• Norman Window Corral
Presentation in Calculus III
• The multivariable corral problem continues without variable substitution
• Maximize enclosed area using “Big D” does not work
• Confirm limitation with a surface plot of the
20 xwwwxxwx fffDandff
wxfA ,
Introduce the Method of Lagrange
Maximize subject to the constraint:
What rectangle has all four sides equal to one-fourth of the perimeter?
ftp 1000xwwxf ),(
022),( pwxwxg impliesgfsogxwf 2,2,
802222),,(,22
pyieldspwxginngsubstitutiwandx
44,
82,2
pwand
pxboth
pandwxSince
The Aviary
• Maximize the volume subject to the constraint of a fixed amount of surface area
• Lagrange Multipliers method or substitution and the use of ‘Big D’
• Proof of the cube as a minimal enclosure
methodDBigforyx
xyxyyxV
methodLagrangeforxyzzyxV
fixedisSxzyzxyS
s
)(),(
),,(
)(2
2
ftzyx
ftSLet
10
600 2
Approximations
• 3-D mesh software (Octave, Matlab) can offer visualizations of maximization
Aviary with n-chambers
• Method of Lagrange n chambers
600)1(22),,(
,,)(
yznnxynxzzyxg
VgnxynxznyzVyznxV
yznnxznxyz
nxyyznnxyz
nxynxznxyz
)1(2)(
2)1()(
22)(
zyyieldspairnd
xn
nyyieldspairst
21
21
planeyzinAreaplanexyinAreaplanexzinArean
nx
n
nx
n
nx
xn
nnx
n
nnxx
n
nnxxg
1
4
1
4
1
4
1
2)1(
1
22
1
22)(
222222
2
Aviary continued
• Aviary with n compartments
• Aviary in the corner of the room
• What do all these problems have in common?
• Conjecture: Any optimal rectangular aviary with any rectangular internal or external additions utilizes equal boundary material in all three dimensions.
2-D or 3-D• What do all the rectangular corrals have in common with the aviaries?
• “Equal boundary material used in xy or xyz directions”
• Sphere has equal material used in all possible directions
• Consider the regular polyhedra in the Isepiphan Problem (Toth,1948)
Double bubble
• Side view is ~1.01 times the area of the top (looking down the longitudinal axis)
• Engineer 10% error – gets promotion
• Physicist 1% error – gets Nobel prize
• Mathematician 1% error – gets back to work
Cube bubble• Boundary conditions are 6
sides in 3-D• Bubbles construct minimal
aviary with the constraint of– Inter wall angle is 120°– Inter edge angle is arc cos(−1/3) ≈
109.4712° (ref: Plateau, 1873)
• Cube angles are nearly 20°or 30° off from Plateau angles
A Little Bubble Lingo• Spherical Bubble that are
joined share walls.
• Edges are where walls and bubbles meet other walls and bubbles
• Three walls/bubbles make an edge
• Edges meet in groups of four (see the end of the straw)
Dodecahedron Bubble• Regular polyhedra
(Platonic Solids) are minimal surfaces for a fixed volume (not fully proven)
• Boundary conditions cause bubbles to create the near-Platonic Solids
• Inter wall and inter edge angles defined by Plateau
• Dodecahedron edge angles are only 7° off from Plateau angles
Tom Noddy on Letterman
Short video
Icosahedron Bubble
• Requires 5 edges to meet (impossible!)
Conclusion
• Have fun with optimization• Have a robust example with
seemingly endless possibilities• Ask students “What is the
overall pattern here?”• Create new problems easily
www.cabrillo.edu/~lsimcik
top related