Transcript chapter 9-9 - Somerset Academy Silver Palms Middle/High

9-9 Scaling Three-Dimensional Figures
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
9-9 Scaling Three-Dimensional Figures
Warm Up
Find the surface area of each rectangular
prism.
1. length 14 cm, width 7 cm, height 7 cm 490 cm2
2. length 30 in., width 6 in., height 21 in 1872 in2
3. length 3 mm, width 6 mm, height 4 mm 108 mm2
4. length 37 in., width 9 in., height 18 in.
2322 in2
9-9 Scaling Three-Dimensional Figures
Problem of the Day
A model of a solid-steel machine tool is
built to a scale of 1 cm = 10 cm. The
real object will weigh 2500 grams. How
much does the model, also made of
solid steel, weigh?
2.5 g
9-9 Scaling Three-Dimensional Figures
Sunshine State Standards
Prep for MA.8.G.5.1 …Convert units of
measure between different measurement
systems…and dimensions
including…area…and derived units to solve
problems.
Rev MA.7.G.2.1
9-9 Scaling Three-Dimensional Figures
Vocabulary
capacity
9-9 Scaling Three-Dimensional Figures
9-9 Scaling Three-Dimensional Figures
Corresponding edge lengths of any two cubes
are in proportion to each other because the
cubes are similar. However, volumes and
surface areas do not have the same scale
factor as edge lengths.
Each edge of the 2 ft cube is 2 times as long
as each edge of the 1 ft cube. However, the
cube’s volume, or capacity, is 23 = 8 times as
large, and its surface area is 4 times as large
as the 1 ft cube’s.
9-9 Scaling Three-Dimensional Figures
Helpful Hint
Multiplying the linear dimensions of a solid by n
creates n2 as much surface area and n3 as much
volume.
9-9 Scaling Three-Dimensional Figures
Additional Example 1A: Scaling Models That Are
Cubes
A 3 cm cube is built from small cubes, each 1
cm on an edge. Compare the following values.
the edge lengths of the two cubes
3 cm cube
1 cm cube
3 cm = 3 Ratio of corresponding
1 cm
edges
The length of the edges of the larger cube is 3
times the length of the edges of the smaller cube.
9-9 Scaling Three-Dimensional Figures
Additional Example 1B: Scaling Models That Are
Cubes
A 3 cm cube is built from small cubes, each 1
cm on an edge. Compare the following values.
the surface areas of the two cubes
3 cm cube
1 cm cube
54 cm2 = 9 Ratio of corresponding
areas
6 cm2
The surface area of the large cube is 9 times that
of the small cube.
9-9 Scaling Three-Dimensional Figures
Additional Example 1C: Scaling Models That Are
Cubes
A 3 cm cube is built from small cubes, each 1
cm on an edge. Compare the following values.
the volumes of the two cubes
3 cm cube
1 cm cube
27 cm3 = 27 Ratio of corresponding
volumes
1 cm3
The volume of the large cube is 27 times that of
the smaller cube.
9-9 Scaling Three-Dimensional Figures
Check It Out: Example 1A
An 8 cm cube is built from small cubes, each 2
cm on an edge. Compare the following values.
the edge lengths of the two cubes
8 cm cube
2 cm cube
8 cm = 4 Ratio of corresponding
2 cm
edges
The edges of the large cube are 4 times as long
as the edges of the small cube.
9-9 Scaling Three-Dimensional Figures
Check It Out: Example 1B
A 8 cm cube is built from small cubes, each 2
cm on an edge. Compare the following values.
the surface areas of the two cubes
8 cm cube
2 cm cube
6(8 cm)2 = 380 cm2 = 16
24 cm2
6(2 cm)2
Ratio of corresponding
areas
The surface area of the large cube is 42 = 16
times that of the small cube.
9-9 Scaling Three-Dimensional Figures
Check It Out: Example 1C
A 8 cm cube is built from small cubes, each 2
cm on an edge. Compare the following values.
the volumes of the two cubes
8 cm cube
2 cm cube
(8 cm)3
(2 cm)3
512 cm3 = 64
8 cm3
Ratio of corresponding
volumes
The volume of the large cube is 43 = 64 times
that of the small cube.
9-9 Scaling Three-Dimensional Figures
Additional Example 2A: Scaling Models That Are
Other Solid Figures
A box is in the shape of a rectangular prism.
The box is 4 ft tall, and its base has a length of
3 ft and a width of 2 ft. For a 6 in. tall model of
the box, find the following.
What is the scale factor of the model?
6 in. = 6 in. = 1
4 ft
48 in. 8
Convert and simplify.
1
The scale factor of the model is 8 .
9-9 Scaling Three-Dimensional Figures
Additional Example 2B: Scaling Models That Are
Other Solid Figures
A box is in the shape of a rectangular prism.
The box is 4 ft tall, and its base has a length of
3 ft and a width of 2 ft. For a 6 in. tall model of
the box, find the following.
What are the length and the width of the
model?
 3 ft = 36 in. = 41 in.
Length: 1
8
8
2
Width:
1
8

2 ft = 24
8 in. = 3 in.
The length of the model is 4 1
2 in., and the width
is 3 in.
9-9 Scaling Three-Dimensional Figures
Check It Out: Example 2A
A box is in the shape of a rectangular prism.
The box is 5 ft tall, and its base has a length of
6 ft and a width of 4 ft. For a 6 in. tall model of
the box, find the following.
the scale of the model?
6 in.
5 ft
6 in. = 1
60 in. 10
The scale of the model is 1:10.
9-9 Scaling Three-Dimensional Figures
Check It Out: Example 2B
A box is in the shape of a rectangular prism. The
box is 5 ft tall, and its base has a length of 6 ft
and a width of 4 ft. For a 6 in. tall model of the
box, find the following.
the length and width of the model?
Length: 1  6 ft = 1 in. 72 ft = 17 1 in.
10
10
2
1  4 ft = 1 in.  48 ft = 4 4 in.
Width: 10
5
10
9-9 Scaling Three-Dimensional Figures
Additional Example 3: Business Application
It takes 30 seconds for a pump to fill a cubic
container whose edge measures 1 ft. How long
does it take for the pump to fill a cubic container
whose edge measures 2 ft?
V = 2 ft

2 ft

2 ft = 8 ft3
Find the volume of the 2
ft cubic container.
Set up a proportion and solve.
30 s
x
=
Cancel units.
3
3
1 ft
8 ft
30

8=x
240 = x
Multiply.
Calculate the fill time.
It takes 240 seconds, or 4 minutes, to fill the larger
container.
9-9 Scaling Three-Dimensional Figures
Check It Out: Example 3
It takes 8 s for a machine to fill a cubic box
whose edge measures 4 cm. How long would it
take to fill a cubic box whose edge measures 10
cm?
Vsmaller box = 4 cm

Vlarger box = 10 cm
4 cm


4 cm = 64 cm3
10 cm

10 cm = 1000 cm3
8s
xs
8000
=
; 8000 = 64x, so x =
= 125
3
3
64
64 cm
1000 cm
It would take 125 seconds, or 2 minutes 5 seconds, to fill.
9-9 Scaling Three-Dimensional Figures
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
9-9 Scaling Three-Dimensional Figures
Lesson Quiz: Part I
A 10 cm cube is built from small cubes, each 1
cm on an edge. Compare the following values.
1. the edge lengths of the two cubes
10:1
2. the surface areas of the two cubes
100:1
3. the volumes of the two cubes
1000:1
9-9 Scaling Three-Dimensional Figures
Lesson Quiz: Part II
4. A pyramid has a square base measuring 185 m on
each side and a height of 115 m. A model of it has
a base 37 cm on each side. What is the height of
the model?
23 cm
5. A cement truck is pouring cement for a new 4 in.
thick driveway. The driveway is 90 ft long and 20 ft
wide. How long will it take the truck to pour the
cement if it releases 10 ft3 of cement per minute?
60 min
9-9 Scaling Three-Dimensional Figures
Lesson Quiz for Student Response Systems
1. A 12 cm cube is built from small cubes, each 3
cm on an edge. Compare the edge lengths of the
two cubes.
A. 12:1
B. 6:1
C. 4:1
D. 3:1
9-9 Scaling Three-Dimensional Figures
Lesson Quiz for Student Response Systems
2. A 20 cm cube is built from small cubes, each 5
cm on an edge. Compare the surface areas of the
two cubes.
A. 15:1
B. 16:1
C. 17:1
D. 18:1
9-9 Scaling Three-Dimensional Figures
Lesson Quiz for Student Response Systems
3. The dimensions of a building are 140 m long,
125 m wide, and 200 m high. The scale model
used to build the building is 14 cm long. What is
the height of the model?
A. 12.5 cm
B. 20 cm
C. 125 cm
D. 200 cm
9-9 Scaling Three-Dimensional Figures
Lesson Quiz for Student Response Systems
4. An aquarium has dimensions 5 ft long, 4 ft wide,
and 6 ft deep. How long will it take to fill the
aquarium with water from a pipe which releases 2
ft3 of water per minute?
A. 15 min
B. 30 min
C. 45 min
D. 60 min