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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
Variations
• Two animals• Three animals
• Two animals by the river• Three animals by the
river
• Is there a pattern in all 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
Corrals of Infinite Internal Complexity
• Infinite number of internal walls, Zeno’s Paradox, for example
perimeterconstPwx
ii
ii
.
11
PwxtoSubject
xwwxAMaximize
:
),(
Corrals of Infinite Internal Complexity
• Substituting out ‘w’ ……leading to:
22
....
1)(
2
P
P
x
and
xP
x
xxPxA
v
….leading to
.
,
2
2
direction
orthogonaleachinused
fencetheofhalfagainonce
Pw
Px
Regular Polyhedra as Optimal Enclosures
• Well known that as the number of sides approaches infinity, the limit shape will be a circle.
2'
221 2
sinlim rn
nrHL
n
Proof that regular polygons are optimal n-sided area enclosures
• Less known: why is a regular n-sided polygon optimal over all other n-gons?
• Convex / concave? Flip out concave portion to prove by contradiction that convex is necessary to be optimal.
• Consider two neighboring sides of the optimal convex n-gon:
Maximize outer triangle area
• Using Heron’s formula:
xccbbcb
xccb
xcbcb
dx
dA
thatshowneasilybecanitand
bcb
xccb
xcbcb
xA
cbbxcxs
24
1
22222
1
,...
2222)(
22
)(
22
1
Each adjacent central triangle is optimal
• Continuing
• And the drawing becomes:
• Conclusions: the outer triangle is necessarily isosceles.
• This is true for all adjacent sides (i.e. adjacent sides are always equal in length)
• Convex n-gon with equal sides is a regular n-sided polygon
2
,,02,024
10 2 c
xsoandxcsoxccbthatimpliesdx
dA
Did you know……
Sides Name
n regular n-gon
3 equilateral triangle
4 square
5 regular pentagon
6 regular hexagon
7 regular heptagon
8 regular octagon
9 regular nonagon
10 regular decagon
But maybe you didn’t know….
Sides Name
11 regular hendecagon
12 regular dodecagon
13 regular triskaidecagon
14 regular tetradecagon
15 regular pentadecagon
16 regular hexadecagon
17 regular heptadecagon
18 regular octadecagon
19 regular enneadecagon
20 regular icosagon
Perhaps don’t want to know
Sides Name
100 regular hectagon
1000 regular chiliagon
10000 regular myriagon
1,000,000 regular megagon
Presentation in Calculus III – returning to the rectangular corral problem
• Rectangular corral with constrained length of fence (say 1000 feet)
• Perimeter equation
• Area Equation is a multi-variable function
xwwxfA ,
100022 wx
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
Moving to 3-D: the Aviary
Assuming fabric on all sides, including the floor
600 square feet of netting, findMaximal volume aviary
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
Aviary with n-chambers
• Method of Lagrange n chambers
600)1(22),,(
,,)(
yznnxynxzzyxg
gVnxynxznyzVyznxV
))1(2(
)2)1((
)22(
yznnxznxyz
nxyyznnxyz
nxynxznxyz
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 n-dimensional rectangular aviary with finite or infinite rectangular internal or external additions (that exists !!!), 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
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