Scale Factors
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Transcript of Scale Factors
![Page 1: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/1.jpg)
Discovering the effect on the volume of a prism when its dimensions are multiplied by a scale factor
Scale Factors and Volume
![Page 2: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/2.jpg)
Find the volume of each prism
1.
38cm9cm
15cm
2.
31,848V m
11m
14m
24m
35,130V cm
![Page 3: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/3.jpg)
Objective
• Given a prism with its dimensions multiplied by a scale factor, you will find the NEW volume without using a formula.
![Page 4: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/4.jpg)
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.
![Page 5: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/5.jpg)
Suppose the length, width, and height of this container were doubled.
We call this “multiplying by a scale factor
of 2.”
![Page 6: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/6.jpg)
Suppose the length, width, and height of this container were tripled.
We call this “multiplying by a scale factor
of 3.”
![Page 7: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/7.jpg)
What is a scale factor?
• In your own words, give a description of a scale factor.
• Share your description with your partner.
![Page 8: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/8.jpg)
How many cubes do you believe would fill this container if the length, width, and height were doubled?
![Page 9: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/9.jpg)
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?
![Page 10: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/10.jpg)
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
![Page 11: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/11.jpg)
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
![Page 12: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/12.jpg)
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
80
Let’s complete our table
![Page 13: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/13.jpg)
We will multiply our original dimensions by a scale factor of 2 and complete our table.
2
2w
2h
8w8
2
w
w
8
2 8 2
w
w
8
2 68 2 1
w
w
![Page 14: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/14.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80
x 3
x 4
x 5
16
Let’s complete our table
![Page 15: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/15.jpg)
We will multiply our original dimensions by a scale factor of 2 and complete our table.
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
![Page 16: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/16.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16
x 3
x 4
x 5
17.2
Let’s complete our table
![Page 17: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/17.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
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
Let’s complete our table
![Page 18: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/18.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
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
Let’s complete our table
![Page 19: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/19.jpg)
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
![Page 20: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/20.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
x 3
x 4
x 5
120
Let’s complete our table
![Page 21: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/21.jpg)
8
3
w
w
8w8
3 8 3
w
w
8
3 48 3 2
w
w
3
3w
3h
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
![Page 22: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/22.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
x 3 120
x 4
x 5
24
Let’s complete our table
![Page 23: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/23.jpg)
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
Now, we will multiply our original dimensions by a scale factor of 3 and complete our table.
![Page 24: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/24.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
x 3 120 24
x 4
x 5
25.8
Let’s complete our table
![Page 25: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/25.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
x 3 120 24 25.8
x 4
x 5
74,304
Find the volumeV=(120)(24)(25.8)
V=74,304V=74,304
Let’s complete our table
![Page 26: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/26.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
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
Let’s complete our table
![Page 27: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/27.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
x 3 120 24 25.8 74,304 ft3 33 or 27 times larger
x 4
x 5
Do you see a pattern? Try completing
the next row on your own.
Let’s complete our table
![Page 28: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/28.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
x 3 120 24 25.8 74,304 ft3 33 or 27 times larger
x 4 160 32 34.4 176,128 ft3 43 or 64 times larger
x 5
Maybe you should
complete the last row.
What can you
conclude?
Let’s complete our table
![Page 29: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/29.jpg)
Length Width Height Volume Volume compared to
original
Original 40 8 8.6 2,752 ft3 SAME
x 2 80 16 17.2 22,016 ft3 23 or 8 times larger
x 3 120 24 25.8 74,304 ft3 33 or 27 times larger
x 4 160 32 34.4 176,128 ft3 43 or 64 times larger
x 5 200 40 43 344,000 ft3 53 or 125 times larger
Last clue.What do you notice about
these two columns?
Let’s complete our table
![Page 30: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/30.jpg)
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
![Page 31: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/31.jpg)
True or False?
2. Whenever the length, width, and height are multiplied by a scale factor of 3, the original volume is multiplied by 34?
True False
or
![Page 32: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/32.jpg)
True or False?
3. Whenever the length, width, and height are multiplied by a scale factor of 23, the original volume is multiplied by 232?
True False
or
![Page 33: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/33.jpg)
True or False?
4. Whenever the length, width, and height are multiplied by a scale factor of 54, the original volume is multiplied by 543?
True False
or
![Page 34: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/34.jpg)
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
![Page 35: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/35.jpg)
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
![Page 36: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/36.jpg)
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
![Page 37: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/37.jpg)
We will multiply our original dimensions by a scale factor of 2 and complete our table.
2
2
6 ft
4 ft 14 ft5 ft 5 ft
2
![Page 38: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/38.jpg)
Let’s complete our 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
12 288
![Page 39: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/39.jpg)
Let’s complete our 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 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
![Page 40: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/40.jpg)
Let’s complete our 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 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
![Page 41: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/41.jpg)
We will multiply our original dimensions by a scale factor of 3 and complete our table.
3
3
6 ft
4 ft 14 ft5 ft 5 ft
3
![Page 42: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/42.jpg)
Let’s complete our 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 12 8 28 1,344 ft3 23 or 8 times larger
x 3
x 4
x 5
18 4212
![Page 43: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/43.jpg)
Let’s complete our 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 12 8 28 1,344 ft3 23 or 8 times larger
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
![Page 44: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/44.jpg)
Let’s complete our 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 12 8 28 1,344 ft3 23 or 8 times larger
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
![Page 45: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/45.jpg)
Let’s complete our 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 12 8 28 1,344 ft3 23 or 8 times larger
x 3 18 12 42 4,536 ft3 33 or 27 times larger
x 4
x 5
By George, I think you’ve got it! Keep
going!
![Page 46: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/46.jpg)
Let’s complete our 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 12 8 28 1,344 ft3 23 or 8 times larger
x 3 18 12 42 4,536 ft3 33 or 27 times larger
x 4 24 16 56 10,752 ft3 43 or 64 times larger
x 5
Go ahead and complete the
last row.
![Page 47: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/47.jpg)
Let’s complete our 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 12 8 28 1,344 ft3 23 or 8 times larger
x 3 18 12 42 4,536 ft3 33 or 27 times larger
x 4 24 16 56 10,752 ft3 43 or 64 times larger
x 5 30 20 70 21,000 ft3 53 or 125 times larger
Did it work? Can we say
that this pattern is true for all prisms?
![Page 48: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/48.jpg)
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
![Page 49: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/49.jpg)
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
![Page 50: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/50.jpg)
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
![Page 51: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/51.jpg)
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
![Page 52: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/52.jpg)
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
![Page 53: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/53.jpg)
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
![Page 54: Scale Factors](https://reader037.fdocuments.net/reader037/viewer/2022102501/54655a69b4af9f354b8b4830/html5/thumbnails/54.jpg)
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