# 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
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:
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
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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