MF9SB CH04 p152-195 pp8 - Mr. Mubashir

NEL152

MF9SB_CH04_p152-195 pp8.qxd 4/7/09 7:46 AM Page 152

GOALS

You will be able to

• visualize composite 3-D objects to reasonabout their surface areas

• solve problems involving surface areas ofprisms and cylinders using reasoning

4Chapter

Measurement

NEL 153

The Royal Canadian Mint inWinnipeg, MB, has a complexshape. How could you estimatehow much glass was used to coverthe building?


CHAPTER 4 Getting Started

Comparing Surface Areas

NEL154 Chapter 4 Measurement

Jarrod and Tiffany make custom candy packages for special occasions.They are working with four new tin designs. Each tin holds about

of candies.

Which tin uses the least amount of material??

24 cm

22 cm18 cm

21 cm

22 cm

24 cm

36 cm

24 cm

24 cm 38 cm

24 cm

9500 cm3


NEL 155Getting Started

surface area

the total area of all the facesof any 3-D object; e.g., thesurface area of this rectangularprism is 2(4 3 6) 5 88 cm2.

2(2 3 4) 1 2(2 3 6) 1

A. The net of the rectangular prism is shown. Draw the nets for theother three tins.

B. Use the nets to calculate the surface area of each tin. Which tinuses the least material?

C. What other factors might Jarrod and Tiffany consider whendeciding which tin to choose?

WHAT DO You Think?Decide whether you agree or disagree with each statement.Explain your decision.

1. Two containers with the same surface area must have the samevolume.

2. To calculate the surface area of this object, break it down into twosmaller objects and add their surface areas.

3. The surface area of a rectangular prism is always more than thesurface area of a triangular prism with the same length, width, andheight.

4. The best way to determine the surface area of this object usessubtraction.

22 cm

18 cm

24 cm

6 cm2 cm

4 cm



Exploring Cube Structures4.1

Build non-prism cube structures and determine their surfaceareas.

GOAL

Shelby designs mirrored tables in various artistic shapes.

She builds each table using cubes made from plywood, and thenattaches mirrors to every exposed surface. These tables are all builtusing eight plywood cubes.

Build four more eight-cube table designs.

Of all the possible designs using eight cubes, which one usesthe most mirrors and which one uses the fewest mirrors?

?

YOU WILL NEED

• linking cubes


Composite Objects andTheir Components

4.2

Decompose objects into right rectangular prisms, right triangularprisms, and right cylinders.

GOAL

LEARN ABOUT the MathYvonne wants to build a set of carpeted pet stairs, as shown. Yvonne couldmake this composite object using rectangular prisms or triangular prisms.

What are the dimensions of the components??

width = 60 cm

length = 80 cm

height = 72 cm

rise

tread

NEL 1574.2 Composite Objects and Their Components

composite object

an object that can bedecomposed into componentparts, such as prisms andcylinders; e.g., this compositeobject can be decomposedinto two rectangular prismsand one triangular prism.

Communication Tip

I saw that I could decompose the stairs into fourrectangular prisms.

EXAMPLE 1 Using different sizes of an object

Nicole’s Solution: Using rectangular prisms

I determined the rise of each step.I assumed that each rise was the same.The shortest prism has a height equal to the rise of the firststep, or 18 cm.Each prism should be 18 cm taller than the previous one.rise of each step 5 72 4 4 5 18 cm

height of 4 steps 5 72 cm

In this book, when you areasked to decompose anobject, it means todecompose it into triangularprisms, rectangular prisms,or cylinders.



60 cm

72 cm

80 cm

18 cm60 cm

20 cm

Looking at the side view, Isaw that I could decomposethe steps to form a largeright triangular prism andsome smaller ones.The large prism is 80 cm long,60 cm wide, and 72 cm high.

I assumed that all of the stepson the stairs are the same size.Each step must have a rise of of the total height, and a tread of of the total length.The width of each step is 60 cm.

14

14

cm cm 5 20 5 18tread 5

14 3 80 rise 5

14 3 72

20 cm 20 cm 20 cm 20 cm60 cm60 cm60 cm60 cm18 cm

36 cm

54 cm

72 cm

Austin’s Solution: Using right triangular prisms

Reflecting

A. Whose solution did you prefer, and why?

B. Suggest another possible solution.

C. Explain how decomposing a composite object makes it easier todetermine its volume.

Next, I determined the tread ofeach step.tread 5 80 4 4 5 20 cm

length of four steps 5 80 cm

I knew that each step was 60 cm wide, so I could labelall the dimensions in my diagram.


WORK WITH the Math


I drew a top view of the base, with itsdimensions.I noticed that each end is semicircular.

EXAMPLE 2 Determining component parts

An aquarium has this object for a base. What component parts make up this base and what are their dimensions?

Derek’s Solution

60 cm

150 cm

top view

The width is a diameter, so the radius of thesemicircles is 30 cm.

I subtracted the radius twice from the overalllength to calculate the distance between thetwo semicircles.cm120 2 30 5 90

150 2 30 5 120

150 cm

80 cm

60 cm

I added the new dimensions to my diagram.

I drew the diagram again, decomposing it intotwo semicircles and a rectangle.

The semicircles in my diagram are the tops of half-cylinders in the actual object. I joined themto form a whole cylinder. The rectangle is thetop of a prism.

90 cm

150 cm

radius30 cm

diameter60 cm

top view

top view 90 cm

60 cm

80 cm

60 cm

60 cm

90 cm80 cm

The components are a rectangular prism and two half-cylinders, which are like a single cylinder.


Practising

2. Sketch two different ways to decompose each object. Include dimensions of each part.


In Summary

Key Idea

• It is possible to decompose a composite object in more than one way.

Need to Know

• When decomposing a composite object, look for the followingcomponent parts: triangular prisms, rectangular prisms, and cylinders.

Checking

1. Sketch two different ways to decompose each object. Include dimensions of each part.a) b)

30 cm

10 cm

18 cm

24 cm

15 cm9 cm

15 cm

8 cm

7 cm

4 cm

a) b) c)

10 cm

5 cm16 cm

8 cm40 cm

30 cm

50 cm 12 cm

10 cm

10 cm

10 cm 10 cm

10 cm

10 cm3. Decompose each object into two components. Include dimensions

of each part.

a) b) c)12 cm

12 cm

12 cm24 cm

24 cm8 cm

8 cm

12 cm

6 cm

8 cm24 cm

20 cm

18 cm30 cm



4. Sketch one way to decompose the red object at right.

5. William wants to make a set ofbookends, one of which isshown. The triangular-prismsupport is inset by 2 cm on allsides. What are the dimensionsof the components for eachbookend?

6. Multiple choice. A video game involves filling a space using thefollowing four objects. Each individual cube measures 1 unit oneach edge. Which one of these objects cannot be decomposed intotwo prisms, each with two cubes?

7. Multiple choice. What is theleast number of triangular orrectangular prism componentsin this object?A. 3B. 4C. 5D. 6

8. Sketch one way to decompose the picture frame shown at right.

9. Describe one way to decompose this desk.

9 cm

21 cm6 cm

7 cm7 cm

3 cm

18 cm16 cm

4 cm

4 cm

2 cm

20 cm

12.5 cm 20.0 cm

2.0 cm25.0 cm

17.5 cm

A. B. C. D.

0.8 m

0.8 m

0.4 m

top viewside view

1.5 m0.9 m

2.3 m


10. Describe one way to decompose this television stand. Assume thatthe cylindrical posts do not go through the bottom shelf.

11. This chair was made by combiningfoam blocks and covering themwith upholstery.a) Describe three ways in which

the chair might have beenmade from component piecesof foam. State the dimensionsof each component part.

b) Could the chair have been madefrom a single foam block? Explain.

12. Determine the component objects and their dimensions for thisgrain elevator.

13. a) Find an example of a composite object in your school, at home,or in your town. Sketch the object and its component parts.

b) Exchange your example with a classmate. Sketch the componentparts of your classmate’s composite object.

c) Compare your solutions for both objects. Did both of youdecompose them in the same way?

Closing

14. Composite objects can be decomposed in more than one way. Explainwith an example why one way might be preferable to another.

Extending

15. This piece of foam is packed around a circular vase for shipping.What are the component parts and what are their dimensions?

16. Sketch a set of component parts,including their dimensions, for thismodel of a toy.

3 cm

3 cm

10 cm

130 cm55 cm

50 cm

d = 5 cm

1.2 m

1.0 m

1.2 m

0.7 m

0.4 m

0.6 m


3.6 m

3.8 m

3.6 m

24.5 m15.2 m

10.3 m9.7 m

24 cm

24 cm

30 cm

30 mm 110 mm

150 mm

60 mm


NEL 1634.3 Area of Overlap

4.3 Area of Overlap

Determine the area of overlap for the components of 3-D objects.

GOAL

LEARN ABOUT the MathAmanda is building a lamp in her woodworking class, using threedifferent pieces of wood as shown. She will glue the bases of the toptwo pieces to attach them to the piece underneath.

What is the total area of the surfaces that will be glued?

A. Build a model of the lamp by stacking the prisms and thecylinder.

B. Examine the model. Trace around the cylinder onto the smallerrectangular prism, then trace around the smaller prism onto thelarger prism.

C. Label on your diagrams the areas of overlap with their dimensions.

D. What is the total area of overlap?

Reflecting

E. Why might you need to know the area of overlap if you weremaking 100 of these lamps?

F. When you are determining the area of overlap for two objects, doyou have to think about the shapes of both objects?

?

YOU WILL NEED

• two different rectangularprisms

• a cylinder• linking cubes• isometric dot paper

area of overlap

the area covered when twocomponent parts are joined toform a composite object

3 cm

18 cm

20 cm 30 cm

28 cm

8 cm

4 cm



WORK WITH the Math

Decompose this structure into rectangularprisms and determine the total areaof overlap. Each edge of each cube is2 cm long.

EXAMPLE 1 Building and decomposing a model

Shelby’s Solution

I decomposed the structure likethis.I used linking cubes of the samecolour for each prism.

I sketched each prism onisometric paper and shaded oneside of each connecting surface.I counted the shaded surfaces.

12 square faces on the prism surfaces are connected to otherprisms, so the total area of overlap is .12 3 4 5 48 cm2

I knew the face of each cube has an area of 2 3 2 5 4 cm2.



15 mm

45 mm

15 mm

15 mm

15 mm

25 mm

The base of the original object is 120 cm long,and the large rectangular prism is 80 cm long.So the small rectangular prism and thetriangular prism are wide.

120 2 80 5 40 cm

The height of the original object is and theheight of the small rectangular prism is Sothe height of the triangular prism is65 2 40 5 25 cm.

40 cm.65 cm,

Area of overlap:Right side of large Top of small rectangular prism rectangular prism

The total area of overlap is.3250 1 2000 5 5250 cm2

5 2000 cm2 5 3250 cm2 5 40 3 50 5 65 3 50

A 5 l 3 w A 5 l 3 w

I noticed that the small rectangular prism andthe triangle overlap the complete side of thelarge rectangular prism. The triangular prismoverlaps the top side of the small rectangularprism.

Brenda makes custom jewellery from metal cubes that are 15 mmon each side. This design has three cubes, as shown. What is thearea of overlap?

EXAMPLE 3 Determining partial area of overlap

Decompose this object and determine the area of overlap.

Bay’s Solution

I visualized separating the object into tworectangular prisms and a triangular prism. I determined the dimensions of the prismsusing given measurements and calculations.

EXAMPLE 2 Decomposing an object using different prisms

80 cm50 cm

65 cm

120 cm

40 cm

40 cm40 cm

50 cm

40 cm

40 cm

50 cm

65 cm

80 cm

50 cm



15 mm

length =25 mm

25 mm

height =45 mm

15 mm

width =25 mm

top view side view I knew that if I placed two metal cubes sideby side, the total width would be 30 mm.

But the design is 25 mm wide, so I couldcalculate the width of the overlap. Thelength of the design was also 25 mm.

15mm15mm30 mm

The cubes overlap byin both directions.30 2 25 5 5 mm

Each cube overlapped another cube by asquare that is 5 mm by 5 mm. There weretwo identical areas of overlap.

5 50 mm2 Total area of overlap 5 2 3 25

5 25 mm2 Area of overlap per cube 5 (5)2

Speakers in the shape of triangular prisms are attached on either side of a television screen. The entire unit is then connected to a base. Determine the total area of overlap.

Derek’s Solution

First, I decided to determine the area ofoverlap between the television screen and the speakers. The screen is 85 cm high, and 8 cm deep. Its edges are rectangular.There is an area that size on either side ofthe screen. I also wanted to determine thearea where the screen touches the base.The width of the screen is 120 cm.

Area of overlap between screen and speakers

Area of overlap between screen and base

5 960 cm25 120 3 8

5 1360 cm25 2 3 (85 3 8)

5 2 3 (l 3 w)

EXAMPLE 4 Determining area of overlap

85 cm

120 cm

25 cm25 cm

16 cm8 cm

Yvonne’s Solution

Next, I needed to calculate the area ofoverlap between the two speakers and thebase. They overlap in right triangular shapes.The base of each triangle is 25 cm, and theheight is 16 cm.

Area of overlap between base and both speakers

5 400 cm2

5 2 3 Q12(25 3 16)R5 2 3 Q12(b 3 h)R



A washer is to be used with a screw. What is the area of overlap between the washer and the screw head?

Amanda’s Solution

I drew a diagram showing how the washerand screw would line up.

The area of overlap between the washerand the screw head is an outer circle thesize of the screw head, with an inner circlethe size of the hole in the washer removed.

I knew that the outer radius is half of theouter diameter of the screw head.I calculated the radius of the hole in thewasher.

I determined expressions for the area ofthe outer circle (the screw head) and theinner circle (the washer hole).

I need to calculate the area of the orange part in my diagram.

Washer dimensions:

5 3 mm Radius of hole 5 10 2 7Full radius 5 10 mm

Area of screw head Area of washer hole

The area of overlap is about 126 mm2. 8 126 mm2 5 p40

Area of overlap 5 p49 2 p9

5 p95 p495 p(3)25 p(7)25 pr 25 pr 2

EXAMPLE 5 Determining circular area of overlap

I subtracted to determine the area ofoverlap.

5 2720 cm25 1360 1 960 1 400

1 area between base and speakers1 area between screen and base

5 area between screen and speakersTotal area of overlap I added all of the areas of overlap together.

top view

hole in washer

shaft of screw

10 mm

7 mm

washerscrew head



Checking

1. Sketch component parts of each object. Indicate the areas of overlap.

In Summary

Key Ideas

• When two objects are combined, there will be an area of overlap.• The way in which a composite object is decomposed may affect its area

of overlap.

Need to Know

• The area of overlap can be all or part of the face of a component part.

2. Calculate the total area of overlap for the components youidentified in question 1.

Practising

3. This object is decomposed into two rectangular prisms and atriangular prism, as shown. Calculate the area of overlap.

4. Multiple choice. A set of three planters is assembled as shown.What is the area of overlap of the exteriors of the planters?A. B. C. D.

5. Multiple choice. A barbell is set up as shown. What is the total area of overlap of the plates?A.B.C.D. 904 cm2

1232 cm2452 cm23620 cm2

60 cm

90 cm

60 cm

150 cm40 cm

100 cm

4000 cm22500 cm22400 cm22000 cm2

a) b) c) 2 cm

14 cm4 cm3 cm

5 cm

3 cm

15 cm15 cm

8 cm4 cm

4 cm

5 cm

12 cm

10.0 cm3.0 cm

4.0 cm

9 cm7 cm

7 cm

3 cm

6 cm21 cm

d = 28 cm

d = 24 cm



6. Calculate the areas of overlap in this cat toy. Each cylindrical posthas a radius of 3.0 cm.

7. A store display is designed with rectangular prisms, as shown. The top of each prism is a square 0.3 m on each side. Calculate thetotal area of overlap of the prisms.

8. Amanda decides to add a fourth triangular piece of wood to the top of her lamp as shown. How would you determine the area ofoverlap between the new triangular prism and the cylinder?

9. Build a composite structure with linking cubes. Exchange yourobject with a partner. Decompose your partner’s object anddetermine the area of overlap.

10. A Kransekake is a Norwegian wedding cake. It is constructed using cookie rings that increase in diameter from top to bottom.Meaghan wants to bake a Kransekake using six cookie rings. The outer diameters for the cookies are 6.0 cm, 7.0 cm, 8.0 cm, 9.0 cm, 10.0 cm, and 11.0 cm. The inner diameter of each cookiering is 2 cm less than its outer diameter. Calculate the total area of overlap of the cookies when assembled.

18 cm

30 cmtop view 28 cm

8 cm4 cm

3 cm

20 cm

50 cm

3 cm

45 cm

0.3 m0.3 m

0.6 m0.9 m

1.2 m0.3 m

0.3 m0.3 m

cookie ring

Reading Strategy

Predicting

Use sketching and decomposing to help predict andconfirm the area of overlap.



Front view

11. A wall unit is made using objects from a popular video game, asshown. Each object consists of four rectangular prisms that are20 cm long, 20 cm wide, and 28 cm high.

a) This wall unit was built using nine of these objects. Calculatethe area of overlap.

b) Use linking cubes to create your own wall unit with 12 of theseobjects. Calculate the area of overlap.

c) Exchange your design with a partner and calculate the area ofoverlap in your partner’s design.

Closing

12. Is it possible for the overlap when two rectangular prisms touch tobe the same as the overlap when two triangular prisms or twocylinders touch? Explain.

Extending

13. This picture frame is made using four individual frames, as shown.This frame is 34.65 cm high and 33.00 cm long. Each individualframe is 17.80 cm by 12.60 cm, with a window of 14.00 cm by8.80 cm. Also, each individual frame overlaps each other frame

by of the width of the frame. The bottom edges of the two lowerframes are even.

a) Determine the dimensions of the open space between theframes.

b) Determine the area of overlap of the four individual frames.

14. A party cake consists of four layers, each the same height. There aretwo square layers, and two circular layers, as shown. Determine thearea of overlap to the nearest square centimetre.

12

20 cm28 cm20 cm

33.00 cm

34.65 cm

30 cm

30 cm


CHAPTER 4 Mid-Chapter Review

NEL 171Mid-Chapter Review

FREQUENTLY ASKED QuestionsQ: How can you decompose a composite object?

A: Examine the object for component rectangular prisms, triangularprisms, and cylinders. There may be more than one way to do this. For example, this kitchen cabinet can be decomposed into a triangular prism, one square prism, and one rectangular prism. Or, it can be made of a triangular prism and three square prisms.

• See Lesson 4.2, Examples 1and 2.

• Try Mid-Chapter Reviewquestions 1 and 2.

Study Aid

• See Lesson 4.3, Examples 1,2, 3, 4, and 5.

• Try Mid-Chapter Reviewquestions 3, 4, and 5.

Study Aid

1.2 m

1.2 m

0.6 m

0.9 m

0.6 m

30 cm20 cm

10 cm

10 cm

20 cm

16 cm 12 cm10 cm

20 cm30 cm

Q: How can you determine the area of overlap for a compositeobject?

A: Separate the object into component parts and calculate the areasof the faces they share. For example, the areas of overlap for thiscomposite object are a semicircle and a rectangle. The rectangle is 20 cm long .

Total overlap area

8 357 cm28 157 1 2008 3.14 3 (10)2 4 2 1 (20 3 10)

5 pr 2 4 2 1 (lw)

5 area of semicircle 1 area of rectangle

30 cm

40 cm

10 cm

20 cm

20 cm

16 cm 12 cm

10 cm

("122 1 162 5 20)



PracticeLesson 4.2

1. Which components can you identify in the International Space Station?

2. Sketch one way to decompose each object. Include dimensions ofeach part.

a) b)

Lesson 4.3

3. Determine the area of overlap for each object in question 2.4. A child’s toy has four wheels, as shown. Determine the area of

overlap between the wheels and the body of the toy.

5. A child made this toy from building blocks. Determine the area ofoverlap.

80 cm

65 cm

20 cm

10 cm

5 cm10 cm

4 cm

8 cm

12.0 cm4.0 cm

11.0 cm6.0 cm

• cylinder• rectangular prism• triangular prism

12.0 m

3.0 m8.0 m

5.0 m7.0 m

7.5 m


Documents

MF9SB CH04 p152-195 pp8 - Mr. Mubashir