jlgeometry of packing booklet

8/6/2019 JLGeometry of Packing Booklet

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The Geometry of Packing

Mathematics 10 Advanced

Ms. Barnaby

2010-2011

Name: _________________

DUE: March 11, 2011

QUIZ: March 17, 2011


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Table of Contents

Title page.1

Table of Contents..2

Outcomes for Chapter 63

Formulas..4

Types of Prisms..........................................................................5

Regular Polygons.......................................................................5

Definitions..6

Activity 1 Measuring Shapes and Calculating Volume....7Activity 2 Packaging Tennis Balls....8

Activity 3 Working with Cones................................................9

Activity 4 Regular Polygons & Applications.............................10

Activity 5 Investigating Area, Perimeter, and Surface Area.....11

Activity 6 Investigating Economy Rate....................................12

Activity 7 Real World Applications..........................................13


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Outcomes for Chapter 6

Outcome Description of OutcomeA3

Demonstrate an understanding of the role of

irrational numbers in applications.

C36

Explore, determine, and apply relationships between

perimeter and area, and between surface area, and

volume.

D1Determine and apply formulas for perimeter, area,

surface area, and volume.

D5

Apply trigonometric functions to solve problems

involving right triangles, including the use of angles of

elevation.

D10

Determine and apply relationships between the

perimeters and area of similar figures and between

the surface areas and volumes of similar solids.

D11 Explore, discover, and apply properties of maximumarea and volume.

D12 Solve problems using the trigonometric ratios.

D13Demonstrate an understanding of the concepts of

surface area and volume.

E1Explore properties of, and make and test conjectures

about, two- and three- dimensional figures.

E2 Solve problems involving polygons and polyhedra.

E8Use inductive reasoning when observing patterns,

developing properties, and making conjectures.

E9

Use deductive reasoning to construct logical

arguments and be able to determine, when given a

logical argument, if it is valid.


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FORMULAS(http://math.about.com)


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PRISMS

REGULAR POLYGONS


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DEFINITIONS

Perimeter

_____________________________________________________________________________

_____________________________________________________________________________

Area

_____________________________________________________________________________

_____________________________________________________________________________

Volume

_____________________________________________________________________________

_____________________________________________________________________________

Surface Area

_____________________________________________________________________________

_____________________________________________________________________________

Capacity

_____________________________________________________________________________

_____________________________________________________________________________

Prism_____________________________________________________________________________

_____________________________________________________________________________

Regular Polygon

_____________________________________________________________________________

_____________________________________________________________________________

Economy Rate

_____________________________________________________________________________

_____________________________________________________________________________


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ACTIVITY 1

Measuring Shapes and Calculating VolumeOutcomes D1, D13, D7

There are 3 shapes that you will need to find in order to complete this activity. You will need to take the

appropriate measurements according to the volume formulas. Measure all units in centimetres. Using the

appropriate formulas, please calculate the volume for each shape. Please use 4 decimal places for all

calculations until you reach your final answer. Final answers should contain proper units and be rounded tothe nearest hundredths.

CONE (Actual object used: ________________________)

Appropriate Volume Formula [1 point] Proper measurements [2 points]

Calculation of volume using measurements [2 points]

CYLINDER (Actual object used: ________________________)



SQUARE-BASED PYRAMID (Actual object used: ________________________)




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ACTIVITY 2

Packaging Tennis BallsOutcomes D1, D13, D7

Tennis balls have a radius of 3.5 cm. A box, which has a lid, is just big enough to contain 6

tennis balls. The arrangement of the balls in the box is shown below.

Final answers should contain proper units and be rounded to the nearest hundredths.

1. Calculate the volume of the box. [2 points]

2. Find the volume of the empty space in the box when the six balls are in it. [3 points]

3. What percentage of the box is empty once the tennis balls are in it? [1 point]

4. Calculate the surface area of the box and its lid. [2 points]

5. Calculate the surface area and volume of a cylindrical tube that would be just big enough to hold 3tennis balls. [3 points]


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ACTIVITY 3

Working with ConesOutcomes D1, D13, D7

The diagram below represents a measuring scoop used for measuring soap powder for a

washing machine. It has a diameter of 8cm and a height of 10cm.

1. Calculate the height of the powder, in centimetres, when the radius of the soap powder in the scoop is3cm. HINT Similar Triangles should help.[3 points]

2. Assume that the same amount of soap powder was poured into a cylindrical container with the sameheight (10 cm) and a base diameter (8 cm) as the cone. What percentage of the cylinder would the

soap powder fill? [3 points]


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ACTIVITY 4

Regular Polygons & ApplicationsOutcomes E2, E8, D1, E2, E8

1. Keeping symmetry in mind, explain the differences between the following shapes. Use a diagram tosupport your answer.

a) A parallelogram and a rectangle [2 points]

b) A rhombus and a square [2 points]

c) A rhombus and a rectangle [2 points]

2. Calculate the surface area of the regular polygon below. You may find it useful to draw three lines ofsymmetry through the vertices to create 6 equal triangles. [5 points]

3. Calculate the volume of a regular decagonal prism with each base side of the decagon measuring 12cm and the height of the prism measuring 76 cm [10 points]

6m


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ACTIVITY 5

Investigating Area, Perimeter, and Surface AreaOutcomes D1, E2, E8, D5, D12, C36, D11

1. Two factors that influence the cost of building a house are the floor area and the perimeter of thehouse. Suppose you were a contractor and you needed to build a house with a main floor area of 64

square meters.

a) Complete the following table. [3 points]Width (m) Length (m) Perimeter (m)

1 64 130

2 32

b) What perimeter results in the least expensive design? [1 points]

2. Merle uses 27 small cubes of ice to build a large block measuring 3 x 3 x 3. He wonders if this largeblock will melt more slowly than 27 cubes kept separate from each other.

a) How many faces will there be on 27 cubes kept separate? [2 points]

b) If they are made into the large block, how many of those faces are exposed? [2 points]

3. Suppose a chunk of ice contains 20 L of water. What would the best dimensions be to minimize thespeed of melting? [3 points]


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ACTIVITY 6

Investigating Economy RateOutcomes C36, E1, E2, D1, D11, E8

Another way to apply geometry is to calculate the Economy Rate (referred to as ER). Economy Rate is the

ratio of Volume to Surface Area.

AreaSurface

VolumeER =

If this ratio is high, that means the package has more volume than surface area, making it a more economical

design. If it is a low number too much material is being used to make the package, making it less economical.

ER has led to some interesting discussions in the marketplace and has contributed to the miniaturization of

packaging of many products in our stores. This is something to consider when you are buying and wrapping

packages. An excessive amount of packaging is thrown out after birthdays and holidays, much of it going to

the landfill.

1. A typical can of tuna holds about 200mL. What are the most efficient dimensions of a tuna can thatmaximize the economy rate? [5 points]

2. Calculate the volume and surface area of the following 4 objects. Place them in order of mosteconomical to least economical. [15 points]


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6 m

8 m

4 m

8 m

ACTIVITY 7

Real World ApplicationsAll Outcomes

1. A tornado is a narrow funnel shaped cloud similar to a cone. The diameter of this tornado is 82 m and its heightis 860 m. Calculate the volume of the funnel of the tornado. [3 points]

2. A rectangular field is 18 m long and 7 m wide. Calculate the cost of putting grass on the field if sod costs$10.75 m2. [4 points]

3. Calculate the area of material needed to produce the sails for this boat. [4 points]


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4. The pyramids in Egypt have held great mysteries for centuries. They are getting quite old, so it hasbeen decided that a coat of protective varnish might help protect this square-based structure. The

Great Pyramid at Giza measures 146 m in height, and has a base-perimeter of 550 m.

a) Find the volume of the pyramid. [2 points]

b) Find the surface area of the pyramid. [4 points] (HINT You may need Pythagorashelp)

5. Road salt and sand is stored in dome-shape storage facilities. This is because road salt and sand isquite heavy and the shape of the cone prevents unnecessary pressure points for breakage (such as

corners!). The cone-shape evenly distributes the pressure from the weight of the salt or sand, and

makes it easier to load it into the storage area. When you pour salt or sand, it forms a cone!

The base is a cylinder with a height of 1.5m and a diameter of 10m. The total height of the building is

8m.

a) Find the surface area of the storage facility, assuming that the door does not exist. [4 points]

b) Find the volume of the storage facility, again assuming that the door does not exist. [2 points].

jlgeometry of packing booklet

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