Scale Factors

Discovering the effect on the volume of a prism when its dimensions are multiplied by a scale factor

Scale Factors and Volume

Find the volume of each prism

1.

38cm9cm

15cm

2.

31,848V m

11m

14m

24m

35,130V cm

Objective

• Given a prism with its dimensions multiplied by a scale factor, you will find the NEW volume without using a formula.

Purpose

• Architects need to know the effect of scale factors on volume when they make scale models of buildings.

• Gardeners need to know the effect of scale factors on volume when doubling the dimensions of a flowerbed in order to purchase the correct amount of soil.

• Knowing the effect of scale factors on volume is a 7th grade standard.

• There is one item on the CAHSEE related to scale factors and volume.

Suppose the length, width, and height of this container were doubled.

We call this “multiplying by a scale factor

of 2.”

Suppose the length, width, and height of this container were tripled.

We call this “multiplying by a scale factor

of 3.”

What is a scale factor?

• In your own words, give a description of a scale factor.

• Share your description with your partner.

How many cubes do you believe would fill this container if the length, width, and height were doubled?

Let’s Explore…

We know that the volume of a 40 foot container is 2,752 ft3. What do you suppose would happen to the volume if we multiplied the length, width and height by a scale factor of 2? Or by a scale factor of 3? Or by a scale factor of 4?

First, let’s make a table

Length Width Height Volume Volume compared to

original

Original 40 8 8.6 2,752 ft3 SAME

x 2

x 3

x 4

x 5

We will multiply our original dimensions by a scale factor of 2 and complete our table.

2

2w

2h

402

40

40

2 40 2

80

40

2 40 2


original


x 2

x 3

x 4

x 5

80

Let’s complete our table


2

2w

2h

8w8

2

w

w

8

2 8 2

w

w

8

2 68 2 1

w

w


original


x 2 80

x 3

x 4

x 5

16



2

2w

2h

8.6h8.6

2

h

h

8.6

2 8.6 2

h

h

8.6

2 8.6 2 17.2

h

h


original


x 2 80 16

x 3

x 4

x 5

17.2



original


x 2 80 16 17.2

x 3

x 4

x 5

22,016Find the volumeV=(80)(16)(17.2)

V=22,016V=22,016



original


x 2 80 16 17.2 22,016 ft3

x 3

x 4

x 5

Whathappened??

It gotbigger!

Maybeyou should

divide.3

22,016 2,752

8 or 2

23 or 8 times

larger


40

3 40 3

403

40

40

3 4 120 3 0

Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.

3

3w

3h


original


x 2 80 16 17.2 22,016 ft3 23 or 8 times larger

x 3

x 4

x 5

120


8

3

w

w

8w8

3 8 3

w

w

8

3 48 3 2

w

w

3

3w

3h



original



x 3 120

x 4

x 5

24


8.6

3

h

h

8.6

3 8.6 3

h

h

8.6

3 8.6 3 25.8

h

h

8.6h

3

3w

3h



original



x 3 120 24

x 4

x 5

25.8



original



x 3 120 24 25.8

x 4

x 5

74,304

Find the volumeV=(120)(24)(25.8)

V=74,304V=74,304



original



x 3 120 24 25.8 74,304 ft3

x 4

x 5What

happened??

Didn’t Itell youalready?

Need help?

3

74,304 2,752

27 or 3

33 or 27 times

larger



original




x 4

x 5

Do you see a pattern? Try completing

the next row on your own.



original





x 5

Maybe you should

complete the last row.

What can you

conclude?



original





x 5 200 40 43 344,000 ft3 53 or 125 times larger

Last clue.What do you notice about

these two columns?


True or False?

1. Whenever the length, width, and height are multiplied by a scale factor of 2, the original volume is multiplied by 23?

True False

or

True or False?


True False

or

True or False?


True False

or

True or False?


True False

or

True or False?

5. Whenever the length, width, and height are multiplied by a scale factor of x, the original volume is multiplied by x 3?

True False

or

Let’s take a look at triangular prisms…

Will the same pattern work with triangular prisms?

6 ft

4 ft 14 ft5 ft 5 ft

First, let’s make a table

Base of the

triangle

Height of the

triangle

Height of the prism

Volume Volume compared to

original

Original 6 4 14 168 ft3 SAME

x 2

x 3

x 4

x 5


2

2

6 ft

4 ft 14 ft5 ft 5 ft

2


Base of the

triangle

Height of the

triangle

Height of the prism


original


x 2

x 3

x 4

x 5

12 288


Base of the

triangle

Height of the

triangle

Height of the prism


original


x 2 12 8 28

x 3

x 4

x 5

1,344

Find the volumeArea of the base = ½(12)(8)

Area of the base = 48V=(48)(28)

V=1,344V=1,344


Base of the

triangle

Height of the

triangle

Height of the prism


original


x 2 12 8 28 1,344 ft3

x 3

x 4

x 5

Whathappened??

Don’t askme!

Make sure you check your work!

3

1,344 168

8 or 2

23 or 8 times

larger


3

3

6 ft

4 ft 14 ft5 ft 5 ft

3


Base of the

triangle

Height of the

triangle

Height of the prism


original



x 3

x 4

x 5

18 4212


Base of the

triangle

Height of the

triangle

Height of the prism


original



x 3 18 12 42

x 4

x 5

4,536Find the volume

Area of the base = ½(18)(12)Area of the base = 108

V=(108)(42)V=4,536V=4,536


Base of the

triangle

Height of the

triangle

Height of the prism


original



x 3 18 12 42 4,536 ft3

x 4

x 5What

happened??

Sorry, I can’tgive you any more help.

3

4,536 168

27 or 3

33 or 27 times

larger


Base of the

triangle

Height of the

triangle

Height of the prism


original




x 4

x 5

By George, I think you’ve got it! Keep

going!


Base of the

triangle

Height of the

triangle

Height of the prism


original





x 5

Go ahead and complete the

last row.


Base of the

triangle

Height of the

triangle

Height of the prism


original






Did it work? Can we say

that this pattern is true for all prisms?

True or False?

1. The volume of a triangular prism is 100m3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 800m3.

True False

or

True or False?

2. The volume of a triangular prism is 310 ft3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 620 ft3.

True False

or

True or False?

3. The volume of a rectangular prism is 10 km3. If the dimensions are multiplied by a scale factor of 4, the new volume will be 270 km3.

True False

or

True or False?

4. The volume of a rectangular prism is 21 yd3. If the dimensions are multiplied by a scale factor of 3, the new volume will be 567 yd3.

True False

or

True or False?

5. The volume of a rectangular prism is 50 m3. If the dimensions are multiplied by a scale factor of x, the new volume will be 50x3 m3.

True False

or

On your own…

1. Suppose the volume of a rectangular prism is 40 ft3. Find the new volume, if the dimensions are multiplied by a scale factor of 3.

2. Suppose the dimensions of a triangular prism are multiplied by a scale factor of 2 and the NEW volume is 432 cm3, what was the original volume?

3. Suppose the original volume of a triangular prism is 120 m3 and the NEW volume is 3,240 m3, what scale factor were the dimensions multiplied by?

1,080 ft3

54 cm3

3

Let’s wrap it up!

Give a brief explanation of what would happen to the volume of the following prism if the dimensions were multiplied by a scale factor.

19ft

Documents

Scale Factors