WHAT’S IN A VOLUME OR SURFACE AREA? 2TEACHER EDITION

List of Activities for this Unit:

ACTIVITY STRAND DESCRIPTION1 – Hand me the Net ME Nets and surface area of rectangular solids.2 – What’s my Area ME Surface area, chg in dimensions3 – Hide in a Sphere They Can’t

Corner You There!ME Spheres, formulas, chg of dimensions table

4 – Change my Size…Cubes ME Cube Table5 – Change my Size…Rectangular

PrismsME Rectangular Solid Table

6 – Sphere Fun ME Sphere Table7 – The Box-It Company ME Surface area and volume apps8 – Name that Volume ME Determine Volume w/various dimensions9 – Backyard Greenhouse ME Surface Area and Volumes of Cylinders10 – Changes to the Greenhouse ME Find the effect on volume when changing

dimensions11 – The Sand Box ME Determine Volume12 – Practice Problems- MC ME (Prob #46-49) What’s in a Volume

Multiple Choice

Vocabulary: Mathematics and ELL

aquarium excavation prismbases faces radiuscylindrical foundation rectangular prismdetermine greenhouse similar diagrams justify solid diameter linear sphere dimensions manufacturer surface area doubles net triples edge perimeter verticeseffect polyhedron  

COE ConnectionsInstalling a PoolZachary's PyramidSpherical Marbles

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 1 of 27

MATERIALSCalculators

Warm-Ups(in Segmented Extras Folder)

Paper PosersPainters ProblemIce Cream DilemmaA Fishy Story

Essential Questions:

What is a net? How does the change in one linear dimension affect the volume? What is meant by the terms solid, prism, edge, vertex, face and polyhedron? What is a rectangular prism? What is a cube? What is a sphere? What are the appropriate labels when determining area or volume? What is the difference between a 2-dimensional and a 3-dimensional figure? How does one calculate the surface area and volume of a cube, rectangular prism and sphere? How can a conclusion be supported using mathematical information and calculations? How can one of the dimensions of a solid be determined when the volume is known? What would happen if one linear dimension of a cube, rectangular prism or sphere is

changed? How is perimeter of a semi-circle determined? How is the area of a semi-circle determined? What is meant by linear foot? How is volume of a cylinder determined?

Lesson Overview:

Before allowing the students the opportunity to start the activity: access their prior knowledge regarding how to do an investigation. What do you already know regarding determining the volume of a solid and what will affect the volume of a solid. Also what they already know regarding terminology involving solids and determining surface area.

Before allowing the students the opportunity to start the activity: access their prior knowledge with regards the students’ experiences with buying products by the linear foot, building greenhouses, the value of having a greenhouse vs. trying to grow plants without one, and experiences with shopping in a hardware store.

A good warm-up for this activity is Paper Posers, Painters Problem, Ice Cream Dilemma or a Fishy Story.

How is a problem situation decoded so that a person understands what is being asked? Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 2 of 27

What mathematical information should be used to support a particular conclusion? 1 gallon 0.13 ft3

A warm-up involving calculating volume and area of a circle could be used. Focus on ratios for sides, perimeter, area and volume. Numbers 28 and 29 should be done in groups and then discussed as a class after the students

have had time to work on solving the problems. How will the students make their thinking visible? Use resources from your building.

Performance Expectations:

4.5.C Identify missing information that is needed to solve a problem.6.2.D Apply the commutative, associative, and distributive properties, and use the order of

operations to evaluate mathematical expressions.6.4.A Determine the circumference and area of circles.6.4.C Solve single- and multi-step word problems involving the relationships among radius,

diameter, circumference, and area of circles, and verify the solutions.6.4.E Determine the surface area and volume of rectangular prisms using appropriate formulas and

explain why the formulas work.7.3.A Determine the surface area and volume of cylinders using the appropriate formulas and

explain why the formulas work.7.3.C Describe the effect that a change in scale factor on one attribute of a two- or three-

dimensional figure has on other attributes of the figure, such as the side or edge length, perimeter, area, surface area, or volume of a geometric figure.

7.3.D Solve single- and multi-step word problems involving surface area or volume and verify the solutions.

8.4.C Evaluate numerical expressions involving non-negative integer exponents using the laws of exponents and the order of operations.

G.6.D Predict and verify the effect that changing one, two, or three linear dimensions has on perimeter, area, volume, or surface area of two- and three-dimensional figures.

Performance Expectations and Aligned ProblemsChapter 16 “What’s in a Volume or Surface Area?” Subsections:

1-Hand me the Net

2- What’s my Area

3- Hide in a Sphere…

4-Ch-ange my Size…Cubes

5-Ch-ange my Size…Rect-angu-lar

6- Sphere Fun

7-The Box-it Com-pany

8-Name that Vol-ume

9-Back-yard Green-house

10-Ch-anges to the Green-house

11-The Sand box

12-MCProb-lems

Problems Supporting:PE 4.5.C

4, 10, 11 1, 2 2 3, 4 2 - 4

Problems Supporting:PE 6.2.D

10 - 13 6, 7 1, 2 1 1 1 1 – 5 1, 2 1 – 3 1 – 4 1 – 4 3, 4

Chapter 16 “What’s in a Volume or Surface

1-Hand me the Net

2- What’s my Area

3- Hide in a Sphere…

4-Ch-ange my Size…

5-Ch-ange my Size…

6- Sphere Fun

7-The Box-it Com-pany

8-Name that Vol-ume

9-Back-yard Green-house

10-Ch-anges to the Green-

11-The Sand box

12-MCProb-lems

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 3 of 27

Area?” Subsections:

Cubes Rect-angu-lar

house

Problems Supporting:PE 6.4.A ≈ 6.4.C

11 – 4

(partially)

1 – 4 (partiall

y)4

Problems Supporting:PE 6.4.E ≈ 7.3.D

1 – 14 1 – 7 1, 2 1 1 1 1 – 5 1 1 – 4 1 – 4 3, 4 3

Problems Supporting:PE 7.3.A

1 – 4 1 – 4 4

Problems Supporting:PE 7.3.C

1 1 1 3 – 5 1, 3, 4 1, 3, 4

Problems Supporting:PE 8.4.C

1 – 5 1, 2 1 1 1 – 4 1, 3, 4 1, 4 4

Problems Supporting:PE G.6.D

1 1 1 3 – 5 1, 4 3, 4

Assessment: Use the multiple choice and short answer items from Measurement and Geometric Sense that are included in the CD. They can be used as formative and/or summative assessments attached to this lesson or later when the students are being given an overall summative assessment.

1-HAND ME THE NET

A solid is a three dimensional figure. A solid has length, width, and height. If a solid has any flat surfaces they are called faces. The line segment where two faces meet is called an edge. Vertices are the points where the edges meet. (The singular term for vertices is vertex.) If the faces of a solid are all polygons, it is called a polyhedron. Some of the solid figures that you may have studied before

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 4 of 27

are cones, pyramids, rectangular solids, cubes, spheres and cylinders. A rectangular solid is a figure made up of rectangles and/or squares. It has 6 faces. A prism is a special type of polyhedron in which two of the faces are parallel and congruent. The parallel faces are called bases. The surface area of a prism is the sum of the areas of all the faces of the prism. Surface area is important when figuring the amount of material needed to cover packages, furniture, and other solid figures.

When you find the surface area of a solid, you are finding the area of the entire surface. Sometimes it helps to cut the figure apart and lay it flat. This cutting a prism and flattening it out has a shape called a net. Shown below is one possible net for the rectangular prism. If you find the area of each of the six rectangles formed and add them up, you find the surface area of the prism.

Rectangular Prism (solid)Vertex

Edge Net for the rectangular prism

Face

1. What is the length and width of rectangle A? The length is 3cm and the width is 2cm.

2. What is the area of rectangle A? The area is 3cm • 2cm = 6cm2.Show your work using words, numbers and/or diagrams.

3. What other rectangular face has the same area as A? F 4. What is the length and width of rectangle B? The length is 4 cm and the width is 2 cm.

5. What is the area of rectangle B? The area is 4cm • 2cm = 8cm2.

Show your work using words, numbers and/or diagrams.

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 5 of 27

6. What other rectangular face has the same area as rectangle B? E

7. What is the length and width of rectangle C? The length is 4cm and the width is 3cm.

8. What is the area of rectangle C? The area is 4cm • 3cm = 12cm2. Show your work using words, numbers and/or diagrams.

9. What other rectangular face has the same area as rectangle C? D

10. What is the surface area of this rectangular prism (solid)? 52cm2

Show your work using words, numbers and/or diagrams.

2(6cm2) + 2(8cm2) + 2(12cm2) = 12cm2 + 16cm2 + 24cm2 = 52cm2

11. A storage box measures 8 inches by 4 inches by 5 inches. The box is sitting in Myra’s front room. She wants to cover the box with paper.

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 6 of 27

What is the surface area of the box, including the top and bottom? 184in2

Be sure to label your answer.Show your work using words, numbers and/or diagrams.

2(4in•5in) + 2(5in•8in) + 2(4in•8in) = 40in2 + 80in2 + 64in2 = 184in2

12. When displaying books at the book fair, Renee likes to cover the display boxes with pretty fabric. To determine how much fabric she needs, she needs to know the surface area of the display box.

What is the surface area of the display box? 413.34cm2

Be sure to label your answer.Show your work using words, numbers and/or diagrams.

2(8.3cm 8.3cm) + 2(8.3cm 8.3cm) + 2(8.3cm 8.3cm) = 6(8.3cm 8.3cm) = 6(68.89cm2) = 413.34cm2

13. John said that, to find the surface area of the rectangular solid below, he added the following Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 7 of 27

areas: 48 sq. cm + 42 sq. cm + 56 sq. cm + 48 sq. cm + 42 sq. cm + 56 sq. cm. Leroy said that the way that he found the surface area was to add the areas, 48 sq. cm + 42 sq. cm + 56 sq. cm, and multiply this sum by 2. Who was correct? Both are correct; their answers are equivalent.

48cm2 + 42cm2 + 56cm2 + 48cm2 + 42cm2 + 56cm2

48cm2 + 48cm2 + 42cm2 + 42cm2 + 56cm2 + 56cm2

2(48cm2) + 2(42cm2) + 2(56cm2)

2(48cm2 + 42cm2 + 56cm2) = 292cm2

14. A cube is a special kind of rectangular solid. All of the faces have the same area. Here is a cube.

a. If you look at the top of a cube what do you see? A square.

b. If you look at a cube from the side, what 2-dimensional figure do you see? A square.

c. If you look at a cube from the front, what 2-dimensional figure do you see? A square.

d. What is true about all of the 2-dimensional figures that make up the faces of a cube?

They are all squares…6 of them.

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 8 of 27

Applying the commutative & associative properties results in:

By the distributive property:

2-What’s My Area

1. The surface area of a cube is easier to determine (find) than the surface area of other rectangular solids.

Why? All the faces are all squares; 6(area of one face).

Given a cube, what does it mean to find the surface area? Find the area of one face and multiply by 6.

2. The surface area refers to the sum of the areas of the faces of a solid. What is meant by the volume of a cube solid? The amount of space a solid occupies or how many cubic units the solid can contain.

3. Given a cube with edge length of 9 inches, find the surface area and the volume. Be sure to label all drawings and your answer.

Surface area 6(9in)(9in) = 6(81in2) = 486in2 Volume (9in)(9in)(9in) = (9in)3 = 729in3

Show your work using words, numbers and/or diagrams (pictures).

4. Given a cube with edge length e, find the surface area and the volume. Be sure to label all drawings and your answer.

Surface area 6(e)(e) = 6(e2) Volume (e)(e)(e) = e3 Show your work using words, numbers and/or diagrams.

5. A cube has a volume of 125 cubic cm. What is the edge length? (5cm)3 = 125cm3 5cm is the

edge length OR = 5cm

What is the surface area? 6(5cm)(5cm) = 6(25cm2) = 150cm2

Show your work using words, numbers and/or diagrams.

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 9 of 27

Teacher Note:

Students will be expected to do this without a calculator.*Review cubes up to 10.

Teacher Note:

Take some time to discuss the meaning of volume. Emphasize changing units.

6. Given a rectangular prism with length of 5 cm, width of 8 cm, and height of 11 cm. Determine the surface area and the volume. Be sure to label all drawings and your answer.

Surface area 366cm2 Volume 440cm3

Show your work using words, numbers and/or diagrams.

SA = 2({5cm•8cm} + {5cm•11cm} + {8cm•11cm}) = 2(40cm2 + 55cm2 + 88cm2) = 2(183cm2) = 366cm2

V = (5cm)(8cm)(11cm) = 440cm3

7. A rectangular prism has a volume of 120 cubic feet. Give three different combinations of dimensions (measurements) for the prism. Determine the Surface Area using your dimensions. Be sure to label.

a. 1ft x 1ft x 120ft Surface Area: 2({1ft•1ft} + {1ft•120ft} + {1ft•120ft}) = 482ft2

b. 3ft x 4ft x 10ft Surface Area: 2({3ft•4ft} + {3ft•10ft} + {4ft•10ft}) = 164ft2

c. 4ft x 5ft x 6ft Surface Area: 2({4ft•5ft} + {4ft•6ft} + {5ft•6ft}) = 148ft2

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 10 of 27

Teacher Note:

As an extension to this problem, ask which prism was the greatest surface area.

*Answers will vary*

3-Hide in a Sphere… They Can’t Corner You There!

1. Given a sphere (a ball shaped solid) of radius r, the formulas to determine the surface area and volume are:

Surface Area = 2 Volume = 3

a. A sphere has a radius of 15 cm. Determine the surface area and volume. Be sure to label all your answers.

a. Surface area SA = 4 (15cm)2 = 900π cm2 ≈ 2827.43 cm2

b. Volume V = 3 = 4500π cm3 ≈ 14,137.17 cm3

Show your work using words, numbers and/or diagrams.

2. A sphere has a volume of 288π cm3. Determine the radius, the diameter, and the surface area. Be sure to label all your answers.

The radius of a circle is the distance from the center of a circle to the edge of the circle.

The diameter of a circle is the distance from one edge of the circle to the opposite edge of the circle that passes straight through the center of the circle.

a. Radius 6cm

b. Diameter 12cm

c. Surface Area SA = 4 r2 = 4 (6cm)2 = 144 ≈ 452.39cm2

Show your work using words, numbers and/or diagrams.

Volume288 = 4/3 r3

R3 = 216r = 6 cm

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 11 of 27

r

rSurface Area4 r2

4 62

144452.16 sq. cm

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 12 of 27

4-Change My Size…Cubes

1. The cube on the right may help you to complete the table.

Length of edge

Surface Area Volume

Change in each edge

length

NewSurface

Area

Describe surf area change

New Volume

DescribeVolume change

2 cm. 24 cm.2 8cm3Doubles (times 2)length

96 cm 2 4 times larger 64cm3

23 = 8 times larger

3 cm. 54cm2 27cm3Triples

(times 3) length

486cm2 32 = 9 times larger

729cm333 = 27 times larger

5in 150 in 2 125in3 Doubles length 600in2

22 = 4 times larger

1000in323 = 8 times larger

4 m 96m2 64m3Multiplies length by

41536m2

42 = 16 times larger

4096m343 = 64 times larger

2.5 m 37.5m2 15.625m3Multiplies length by

103750m2

100 times larger

(102 = 100)15,625m3

103 = 1000 times larger

K units 6K2units2 K3units3Multiplieslength by

m

m2•6K2

units2m2 times

largerm3•K3

units3m3 times

larger

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 13 of 27

AB

C

5-Change My Size…Rectangular Prisms

1. The rectangular prism on the right may help you to complete the table.

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 14 of 27

A B C SurfaceArea Volume Change in

edges

New Surface

Area

Describe surf area change

New Volume

Describe volume change

A B C SurfaceArea Volume Change in

edges

New Surface

Area

Describe surf area change

New Volume

Describe volume change

4 cm 5 cm 9 cm 202 cm² 180 cm³ Double all sides 808 cm² 4 times as

big (2²) 1440 cm³8 times as big

(2³)

2” 6” 6" 120 in² 72 in3 Multiply all edges by 3 1080 in² 9 times as

big (3²) 1944 in³27 times

as big (3³)

4 m 7 m 8 m 232 in² 224 m3 Multiply A & B by 4 1600 in²

6.89655 times as

big3584 in³

16 times as big

(4²)

6 in 5 in 4 in 148 in2 120 in³ Multiply all edges by 3 1332 in² 9 times as

big (3²) 3240 in³ 27 times as big

2 ft 10 ft 7 ft 208 ft2 140 ft³ Multiply all edges by 4 3328 ft² 16 times

as big (4²) 8960 ft³64 times

as big (43)

6-Sphere Fun1. Use the sphere on the right to help complete the table. (SA = ; V = )

Radius Surface Area Volume Change in

radiusChange in diameter

Change in circumference

Change in area

Change in volume

4 in.64 in² ≈ 201.06in2 85 in3 ≈

268.08in3

Double radius (2 times larger)

2 times larger 2 times larger 22 = 4 times

larger23 = 8 times larger

5.2 in 108.16 in² ≈ 339.79in2

187.48 in³ ≈ 588.98in3 3 times larger

Triple diameter (3 times larger)

3 times larger 32 = 9 times larger

33 = 27 times larger

3.76 cm. 56.5504cm² ≈ 177.66cm2

70.88 cm³ ≈

222.67cm32 times larger 2 times

largerDouble

Circumference (2 times larger)

22 = 4 times larger

23 = 8 times larger

3 in 36 in² ≈ 113.1 in2

36π in3 ≈ 113.1in3 3 times larger 3 times

larger 3 times larger 9 times larger x by 2197

3 m. 36 m² ≈ 113.1 m2

36 m³ ≈ 113.1m3 13 times larger 13 times

larger 13 times larger 132 = 169 times larger

133 = 2197 times larger

K units 4 k²units2 k3units3 20 times larger 20 times larger 20 times larger 202 = 400

times larger203 = 8000

times larger

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 15 of 27

r

7-The Box-it Company

1. The Box-It Company is producing a new design for one of its rectangular cardboard boxes. The area of the top of the box is 192 inches, the area of the front of the box is 192 inches, and the area of the side of the box is 144 inches.

What is the volume of the box? Be sure to label correctly. 2304in3

Show your work using words, numbers and/or diagrams.Factors to get 144: 1x144, 2x72, 3x48, 4x36, 6x24, 8x18, 9x16, 12x12

Factors to get 192: 1x192, 2x96, 3x64, 4x48, 6x32, 8x24, 12x16

Look for factors that you can find throughout all 3 side areas: the sides are 12x12, 12x16, and 12x16

Volume: 12in x 12in x 16in = 2304in3

2. The Box-It Company is producing another new design for one of its rectangular cardboard boxes. The area of the top of the box is 192 inches, the area of the front of the box is 108 inches2, and the area of the side of the box is 144 inches.

What is the volume of the box? Be sure to label correctly. 1728in3

Show your work using words, numbers and/or diagrams.Factors to get 108- 1x108, 2x54, 3x36, 4x27, 6x18, 9x12

Factors to get 144- 1x144, 2x72, 3x48, 4x36, 6x24, 8x18, 9x16, 12x12

Factors to get 192- 1x192, 2x96, 3x64, 4x48, 6x32, 8x24, 12x16

Look for factors that you can find throughout all 3 side areas: the sides are 9x12, 9x16, and12x16V = (9in)(12in)(16in) = 1728in3

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 16 of 27

192in2 12in

144in2

192 in2 12in

16in

144in2 9in

108in2

192in2 12in

16in

3. The company decides to double all of the dimensions of the box from problem 2.

a. What happens to the surface area of the box? The SA will be 22 = 4 times larger. So the new surface area is 4• 888in2 = 3552in2.

Show your work using words, numbers and/or diagrams.Original SA = 2({l • w} + {l • h} + {w • h}) = 2 lw rectangles + 2 lh rectangles + 2wh rectangles

= 6 facesDoubling all the dimensions: New SA = 2(2l • 2w +2l • 2h + 2w • 2h) = 2(4{l •w} + 4{l •h} + 4{w •h}) = 4 • 2({l • w} + {l • h} + {w • h}) = 22• Original SA = 4• Original SA

b. What happens to the volume of that box? The volume will be 23 = 8 times larger. So the new volume is 13,824in3

Show your work using words, numbers and/or diagrams.Original V = lwh

Doubling the dimensions:New V = 2l2w2h = 23(lwh) = 8(lwh) = 8• Original V 8• 1728in3 = 13,824in3

4. A firework explodes and is spreading from its center in a spherical pattern. The sphere currently covers a surface area of 240 sq. meters. The diameter of the firework is expected to double in the next 3 seconds. What will be the new surface area of the sphere? 22 = 4 times larger 4(240m2) = 960m2.

Show your work using words, numbers and/or diagrams.

5. A coffee shop is having rectangular doughnut boxes made. The small box is 11” by 17” while the larger one is twice as long and twice as wide, but the same depth. The box-making company charges by the cubic inch for boxes; the cost of the smaller box is $0.45 per box.

How much should they charge for the larger one? 22•$0.45 = 4• $0.45 = $1.80 Show your work using words, numbers and/or diagrams.

Because the depth is constant the only material changes to make the box is the change in length and the change in width as shown. The area is increased by a factor of four; it follows that the cost would increase by a factor of four also.

Twice as long and twice as wide 4 times the cost.

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 17 of 27

8-Name that Volume1. Marni wants an aquarium (fish tank) that measures 24 inches by 18 inches by 15 inches. The

empty tank weighs 7.4 lbs. Marni knows that 1 gallon of water weighs 8.345 lbs. What will be the total weight of the aquarium when filled with water ? (1 gal = 0.13 ft.3) 228.44 lbs.Show your work using words, numbers and/or diagrams.

Vaquarim = (24in)(18in)(15in) = 6480in3 • =

Weight of the water = ( )( )( ) = 240.72 lbs.

Weight of the tank and the water = 7.4 lbs + 240.72 lbs = 248.12 lbs.

-OR-

Weight of tank and water = 7.4 lbs + •( )•( ) = 248.12 lbs.

2. The diagram shows the ground plan of a mini shopping mall. The excavation (digging) for its foundation (base) is to be 5 feet in depth.

200ft

Area = (200ft)2

= 40,000ft2

Area = (400ft) (300ft) =120,000ft2

Area = (250ft)(480ft) = 120,000ft2

How many cubic yards of soil must be removed? 51,851.85yd3 Show your work using words, numbers and/or diagrams.

Volume to be removed in feet cubed = (total area)(the depth 5ft) = (120,000ft2 + 40,000ft2 + 120,000ft2)•(5ft)

= (280,000ft2)•(5ft) = 1,400,000ft3

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 18 of 27

400 ft.

300 ft.

100 ft.

200 ft.

280 ft.

250 ft.

480 ft.

450 ft.

100 ft.

Volume to be removed in feet yards = 1,400,000ft3•( ) = 51,851.85yd3

9-BACKYARD GREENHOUSE

1. Use the cylinder to help complete the table. (SA = 2 π r2 + 2 π r h; V = π r2 h)

Radius (r) Height (h) Surface Area (SA) Volume (V)8 feet 10 feet 904.78ft2 2010.62ft3

12 in. 20in 2412.7432 in2 9047.79in3

3.5 cm. 6 cm. 208.92cm2 230.91cm3

5 m. 6m 471.24m2 150 m3

Radius, r

Height, h

2. The backyard greenhouse (plant farm) in the figure uses plastic tubing for framing and plastic sheeting for wall covering. The end walls are semicircles, and the greenhouse is built to the dimensions in the figure. All walls and the floor are covered. The door is formed by cutting a slit in one of the end walls.

Width LengthHow many square feet of plastic sheeting will it take to cover top and ends of the greenhouse?

It will take: 314.16ft2 + 78.54ft2 = 392.70ft2 of plastic sheeting.

Show your work using words, numbers and or diagrams.

a. Top: The top is half the lateral area of a cylinder = (0.5)(2πr•h) = (0.5)(2 5ft•20ft) = 314.16ft2

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 19 of 27

b. Ends: The ends are two half circles; Area circle = r2 = (5ft)2 = 78.54ft2

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 20 of 27

3. What is the volume of the greenhouse? 785.40ft3

Show your work using words, numbers and or diagrams.

The Greenhouse is half the volume of a cylinder with the same dimensions.

V = (0.5)( r2h) = (0.5)( )(5ft)2(20ft) = 785.40ft3

4. What happens to the volume of the greenhouse if the floor’s width is tripled? r = 3(5ft)

It will take: 942.48ft2+ 706.86ft2= 1649.34ft2 of plastic sheeting.

The top is half the lateral area of a cylinder = (0.5)(2πr•h) = (0.5)(2 15ft•20ft) = 942.48ft2

The ends are two half circles; Area circle = r2 = (15ft)2 = 706.86ft2

The volume would be: V = (0.5) r2h = (0.5) (15ft)2(20ft) = 7,068.58ft3

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 21 of 27

10-Changes to the Greenhouse

1. The person building the greenhouse discovered that he did not have enough room to build one with the original dimensions. He decided that all dimensions need to be cut in half.What effect does that have on the original volume? The volume is one-eighth the original.Show your work using words, numbers and/or diagrams.

Voriginal = r2h Vnew = ( )2• = = ( ) r2h = ( )• Voriginal

2. The plastic sheeting used for this greenhouse is sold in widths of 10 feet and costs $0.96 per linear (straight) foot. What would be the total cost for covering the original greenhouse? Show your work using words, numbers and/or diagrams.

1foot by 10 feet 10ft2 for $0.96

It will take: 314.16ft2 + 78.54ft2 = 392.70ft2 of plastic sheeting.

(392.70ft2 of plastic sheeting)( ) = $37.70

3. What would happen to the original cost for the plastic sheeting if the length of the floor dimensions were doubled?

Show your work using words, numbers and/or diagrams. 2•rold = 10ft, 2•lold = 40ft

SAold = (2 •rold + (0.5) (rold)2•lold )

SAnew = (2 {2•rold}•{ 2•lold } + {2•rold}2•{2•lold }) = 4({2 •rold } + {(0.5) (rold)2•(lold )}) = 4•SAold

The new SA will be 4 times larger!

Vold = 0.5 r2h, Vnew = (0.5) (2r)2(2h) = 8(0.5)( r2h) = 8•Vold

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 22 of 27

10ft

The new volume will be 8 times larger!

4. Dwayne was creating square posters for class and found some interesting relationships between side length and area. Dwayne decided to double the length of each side of a square.

Determine the effect (results) on the perimeter (distance around)

Determine the effect on the area

Show your work using words, numbers and/or diagrams.

- When he doubles the length of the sides, the perimeter will also be doubled.

- The area will be multiplied by 4. let s = the side length of the square; (2s)2 = 4s2, so four times the area.

Dwayne decided to triple the length of the rectangle, but left the width the same.

Determine the effect on the perimeter

Determine the effect on the area

Show your work using words, numbers and/or diagrams.

For the perimeter:Let P = perimeter, l = length, and w = width. Old rectangle: P = 2l + 2w. New rectangle: P = 3(2l) + 2w = 6l + 2w.

For the area:Let A = area, l = length, and w = width. Old rectangle: Ao = l•w. New rectangle: An = 3l•w = 3(l•w) = 3A

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 23 of 27

11-The Sandbox

1. Your little brother wants a sand box that is 8 feet by 10 feet by 2 feet. He wants to fill it half full of sand. Sand comes in boxes that are 1 foot by 1 foot by 1 foot or 2 feet by 2 feet by 2 feet.

a. How many small boxes would he need? He would need 80 small boxes.

b. How many large boxes would he need? He would need 10 large boxes.Show your work using words, numbers and/or diagrams.

Small boxes: 8•10•1 = 80 80÷13 = 80

Large boxes: 8•10•1 = 80 80÷23 = 10

c. What is the ratio (fraction) of the number of small boxes needed to the number of large boxes

needed to fill his sandbox? = =

2. Mario’s neighbor used leftover pieces of lumber to build the sandbox shown below. The sandbox is 9 feet by 12 feet.

Mario offers to help her fill the sandbox with sand. The sand they will use comes in 60 pound bags. They want the sand to be 5 inches deep. The sand weighs 95 pounds per cubic foot. How many bags of sand will they need? _________________________________________________Show your work using words, numbers and/or diagrams.

Cubic feet of sand needed: (9ft)(12ft)( ft) = 45ft3.

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 24 of 27

Weight of 45ft3 of sand: ( )(45ft3) = 4275 lbs.

Number of bags of sand needed: (4275 lb)( ) = 71.25 bags 72 bags will be needed.

3. Jessica wants a rectangular backyard swimming pool built in her backyard that is 10 feet wide and 22 feet long. The depth of the pool will be 3 feet at the shallow end, sloping to 7 feet at the deep end. What is the volume of water needed to fill the pool? 1100ft3

Show your work using words, numbers and/or diagrams.

4. Ian wants to have a cylindrical (circular top and bottom) swimming pool with radius 12 feet and depth 7 feet installed in the backyard. How many gallons of water will the pool hold when it is filled to within one foot of the top? (1 gallon 0.13 ft3) 20,879.51 gallonsShow your work using words, numbers and/or diagrams.

12ft

6ft

Area of top: A = r2 = (12ft)2 = 144ft2 ≈ 452.39ft2

Volume in cubic feet: A•(depth of water) = (452.39ft2)(6ft) = 2,714.34ft3

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 25 of 27

22ft 10ft 7ft

3ft

Voulme top rectangular prism: (3ft)(10ft)(22ft) = 660ft3

Volume bottom “wedge”: = 440ft3

Total volume: 660ft3 + 440ft3 = 1100ft3 of water

Volume in gallons: (2,714.34ft3)( ) = 20,879.51 gallons of water needed to fill to a 6

foot depth.

Practice Problems

1. Bonni has two similar (same shape) rectangular boxes. The dimensions of box 1 are two times the dimensions of box 2.

Which number shows how many times greater is the volume of box 1 than the volume of box 2?

A. 3

B. 6

C. 8

D. 9

2. A rectangle that is NOT a square has side lengths that are whole numbers. The perimeter of the rectangle measures 8 cm.

Which is the area of the rectangle?

A. 3 cm²

B. 4 cm²

C. 12 cm²

D. 15 cm²

3. A can manufacturer (maker) plans to print the company name across the bottom of its cylindrical cans and also around the middle of the can. They discovered they could print the name, side by side, 7 times across the diameter of the bottom of each can.

Which is the number of times the name could fit around the middle of the can?

A. 14

B. 22

C. 35

D. 49Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 26 of 27

4. Brittany and Paul both have fish tanks shaped like cubes. Each side of Brittany’s tank is ½ the length of each side of Paul’s tank. Which represents the volume of Paul’s tank in comparison to the volume of Brittany’s tank?

A. 2 times as large

B. 4 times as large

C. 8 times as large

D. 16 times as large

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 27 of 27