CAN YOU BUILD THE BEST BOAT by qtd21362

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									  CAN YOU BUILD
  THE BEST BOAT?

   We’ll Have Supplies;
You Bring the Mathematics!
    Matt Kaufmann & Darl Rassi
        www.gk12.ilstu.edu/presentation/
The Idea
  From a piece of cardboard, square corners
  are cut out so that the sides can be folded up
  to make a box. Express the volume of the
  box as a function of the height.
  The Problem
The boat with the most volume will hold the most mass.

        Build a boat that holds the most mass using…
          •One piece of laminated paper
          •Tape
          •Rulers, scissors, calculators, etc
Development
  Calculus
     •Problem-solving & critical thinking
     •Writing functions
     •Understanding graphs
     •Optimization
  Algebra I
     •Problem-solving & critical thinking
     •Writing expressions to model
     •Multiplying binomial expressions
     •Simplifying expressions
     •Understanding graphs
Algebra I
  Each group:
  •Calculates the footprint (area of the
  bottom), total surface area, volume, and
  perimeter for their boat height (e.g. 0.5”)
  •Build their boat
  •Sink their boat
  •Record the mass the boat held
  •Report the capacity to the teacher/class
  •Each student graphed the data in Excel
  •The data is analyzed
                                     Mass Held Vs Boat Height                                                                                                                                Mass Held Vs Boat Footprint

                450                                                                                                                                                  450

                400                                                                                                                                                  400

                350                                                                                                                                                  350

                300                                                                                                                                                  300




                                                                                                                                                  Mass Held (g)
Mass Held (g)




                250                                                                                                                                                  250

                200                                                                                                                                                  200

                150                                                                                                                                                  150

                100                                                                                                                                                  100

                50                                                                                                                                                   50

                 0                                                                                                                                                    0
                      0       0.5            1                1.5             2                      2.5                                                                   0        5       10     15            20             25        30        35      40
                                                                                                               Mass Held vs Perimeter
                                              Side Height (in)                                                                                                                                      Area of Footprint (in2)

                                                                                          450

                                                                                          400

                                                                                          350

                                                                                          300
                                                                          Mass Held (g)




                                                                                          250

                                                                                          200

                                                                                          150

                                                                                          100
                                                                                          50
                                     Mass Held vs Surface Area                             0                                                                                                     Mass Held vs Volume                 y = 18.627x + 4.9388
                                                                                                0          5    10          15          20   25                                30
                                                                                                                                                                                                                                          R2 = 0.9368
                450                                                                                                    Perimeter (in)                                450
                400                                                                                                                                                  400
                350                                                                                                                                                  350
                300
Mass Held (g)




                                                                                                                                                                     300




                                                                                                                                                     Mass Held (g)
                250                                                                                                                                                  250
                200                                                                                                                                                  200
                150                                                                                                                                                  150

                100                                                                                                                                                  100

                50                                                                                                                                                    50

                 0                                                                                                                                                     0
                      0   5     10      15       20      25         30   35                     40   45                                                                    0            5            10                    15                  20           25
                                                              2
                                             Surface Area (in )                                                                                                                                         Volume of Boat (in 3)
www.bigfoto.com
VOLUME!!!
  We conclude that volume is the most
  important characteristic.
Algebra I:
Need for Variables
   To prepare algebra students for Task 8
   we explore a Sketchpad file
                                        BC =
                      A         B              C
                                                   CD =

        AF = 6.0 cm

                                G              D


                      F                        E
                          FE = 8.0 cm


   GSP File
Algebra I:
Writing Equations
   Now using a second GSP file, students are
   asked to find an expression for the
   footprint of all boats.
   •Then they find expressions for surface
   area, volume, and perimeter of the rim of
   the boat.

     GSP File
Algebra I:
Task 11
  CAN YOU BUILD
  THE BEST BOAT?

   We’ll Have Supplies;
You Bring the Mathematics!
The Challenge:
Algebra II/ PreCalculus
     Build a boat that holds the most mass
     using…
        •One laminated notecard
        •Tape
        •Rulers, scissors, calculators, etc
    Tape can be used to
    •Connect pieces of paper
    •Seal seams
    • But cannot be used to increase surface area!
The Challenge:
Algebra II/ PreCalculus
       Be Mathematical
                      AND

    Think            Outside
        The Box!!!
   … but it still has to be a rectangular prism!
The Best Boat: Optimize!
                            9-2x

                                     x Inches
       5 Inches




                                     5-2x




                          9 Inches
Volume:
   V ( x)  9  2 x 5  2 x x       56  562  41245
                                     x
              4 x3  28x 2  45x                 24
   V '  x   12 x 2  56 x  45      1.03 or 3.64
The Best Boat: Optimize!
Dimensions:
    3” x 7” x 1”
    Volume:
 Is this the BEST Boat?
         21 in3
The Best Boat: Optimize!
 And use all of the paper !!!
    Area originally wasted: 4 in2
    Perimeter: 2(3+7) = 20 in
    Additional Height: 4/20=0.2 in
So… Volume: 3x7x1.2 = 25.2 in
   New                           3

                              vs = 21 in
                                     3

  Is this the BEST Boat?
The Best Boat:
Optimize the entire paper!!!

So…Volume:
                                 
    V ( x)  9  2 x 5  2 x  x 
                                 
                                 
                                       4x2 
                                        p 
                                           
                                           

  Is this the BEST Boat?
  Where
      4x
           
               is the area of the four squares
                   2

  and p  2 9  2 x  5  2 x is the perimeter of the boat
  We find maximum at x≈1.22
  New Volume: 6.56 x 2.56 x 1.55 = 26.0 in3
                                              vs = 25.2 in3
The Best Boat:
        Optimize Design!!!
  By cutting out corners, we will never get a
  SQUARE BASE!
   Square base: length = width = x
So…              height = y
   Surface Area: 45=4xy+x2
                   y=(45-x2)/(4x)
  Is this the BEST Boat?
        
   Volume: V x  x 
                     45  x 2 
                    2
                    4x      
                             
   We find maximum at x≈3.87
   New Volume: 3.87 x 3.87 x 1.94 = 29.0 in3
                                            vs = 26.0 in3
Canoe
Frustum of a Cone
Cylinders
“Rowboat”
“Flair” Boat
Hexagonal Prism
Optimal Box
Square Prism
Half-cylinder Prism
Sphere
Half-ellipse Prism
Questions?
Thank you for attending
   our presentation!
        www.gk12.ilstu.edu/presentation



 Matt Kaufmann              Darl Rassi
kaufmannm@district87.org mr.rassi@gmail.com

								
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