Scale Factors and Volume. Discovering the effect on the volume of a prism when its dimensions are multiplied by a scale factor

Transcription

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

2 Find the volume of each prism cm 14m 11m 24m 38cm 9cm V = 1,848m 3 V = 5,130cm 3

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

4 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 7 th grade standard. There is one item on the CAHSEE related to scale factors and volume.

5 Suppose the length, width, and height of this container were doubled. We call this multiplying by a scale factor of 2.

6 Suppose the length, width, and height of this container were tripled. We call this multiplying by a scale factor of 3.

7 What is a scale factor? In your own words, give a description of a scale factor. Share your description with your partner.

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

9 Let s Explore We know that the volume of a 40 foot container is 2,752 ft 3. 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?

10 First, let s make a table Length Width compared to original Original ,752 ft 3 SAME x 2 x 3 x 4 x 5

11 We will multiply our original dimensions by a scale factor of 2 and complete our table. h 2 l 2 w 2 l = 40 l 2 = 40 2 = 80

12 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x 2 80 x 3 x 4 x 5

13 We will multiply our original dimensions by a scale factor of 2 and complete our table. h 2 l 2 w 2 w = 8 w 2 = 8 2 = 16

14 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x x 3 x 4 x 5

15 We will multiply our original dimensions by a scale factor of 2 and complete our table. h 2 l 2 w 2 h = 8.6 h 2 = = 17.2

16 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x x 3 x 4 x 5

17 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x 2 x 3 x Find the volume V=(80)(16)(17.2) V=22,016 22,016 x 5

18 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x 2 x 3 x Maybe you should divide. 22,016 ft or 8 times It got bigger! 3 22,016 2,752 = What happened?? 8 or 2 x 5

19 Now, we will multiply our original dimensions by a scale factor of 3 and complete our table. h 3 l 3 w 3 l = 40 l 3 = 40 3= 120

20 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x ,016 ft or 8 times x x 4 x 5

21 Now, we will multiply our original dimensions by a scale factor of 3 and complete our table. h 3 l 3 w 3 w = 8 w 3 = 8 3 = 24

22 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x ,016 ft or 8 times x x 4 x 5

23 Now, we will multiply our original dimensions by a scale factor of 3 and complete our table. h 3 l 3 w 3 h = 8.6 h 3 = = 25.8

24 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x ,016 ft or 8 times x x 4 x 5

25 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x ,016 ft or 8 times x ,304 x 4 x 5 Find the volume V=(120)(24)(25.8) V=74,304

26 Let s complete our table Length Width compared to original Original ,752 ft 3 SAME x 2 x 3 x Need help? ,016 ft 3 74,304 ft 3 74,304 2,752 = 2 3 or 8 times Didn t I tell you already? 3 3 or 27 times x 5 What 3 happened?? 27 or 3

27 Let s complete our table Original Length 40 Do Width you see a pattern? Try completing the next row on 8your own ,752 ft 3 compared to original SAME x ,016 ft or 8 times x ,304 ft or 27 times x 4 x 5

28 Let s complete our table Length What Maybe can you Width you should conclude? complete the last row. compared to original Original ,752 ft 3 SAME x ,016 ft or 8 times x ,304 ft or 27 times x ,128 ft or 64 times x 5

29 Let s complete our table Original Length 40 Width Last clue. What do you notice about these two columns? ,752 ft 3 compared to original SAME x ,016 ft or 8 times x ,304 ft or 27 times x ,128 ft or 64 times x ,000 ft or 125 times

30 True or False? 1. Whenever the length, width, and height are multiplied by a scale factor of 2, the original volume is multiplied by 2 3? or True False

31 True or False? 2. Whenever the length, width, and height are multiplied by a scale factor of 3, the original volume is multiplied by 3 4? or True False

32 True or False? 3. Whenever the length, width, and height are multiplied by a scale factor of 23, the original volume is multiplied by 23 2? or True False

33 True or False? 4. Whenever the length, width, and height are multiplied by a scale factor of 54, the original volume is multiplied by 54 3? or True False

34 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? or True False

35 Let s take a look at triangular prisms Will the same pattern work with triangular prisms? 5 ft 5 ft 4 ft 14 ft 6 ft

36 First, let s make a table Base of the triangle of the triangle of the prism compared to original Original ft 3 SAME x 2 x 3 x 4 x 5

37 We will multiply our original dimensions by a scale factor of 2 and complete our table. 2 5 ft 5 ft 4 ft 14 ft 6 ft 2 2

38 Let s complete our table Base of the triangle of the triangle of the prism compared to original Original ft 3 SAME x x 3 x 4 x 5

39 Let s complete our table Base of the triangle of the triangle of the prism compared to original Original ft 3 SAME x ,344 x 3 x 4 x 5 Find the volume Area of the base = ½(12)(8) Area of the base = 48 V=(48)(28) V=1,344

40 Let s complete our table Base of the triangle of the triangle of the prism compared to original Original ft 3 SAME x 2 x 3 x Make sure you check your work! 1,344 ft 3 1, = What happened?? 8 or or 8 times Don t ask me! 3 x 5

41 We will multiply our original dimensions by a scale factor of 3 and complete our table. 3 5 ft 5 ft 4 ft 14 ft 6 ft 3 3

42 Let s complete our table Base of the triangle of the triangle of the prism compared to original Original ft 3 SAME x ,344 ft or 8 times x x 4 x 5

43 Let s complete our table Base of the triangle of the triangle of the prism compared to original Original ft 3 SAME x ,344 ft or 8 times x Find the volume x 4 Area of the base = ½(18)(12) Area of the base = 108 x 5 V=(108)(42) V=4,536 4,536

44 Let s complete our table Base of the triangle of the triangle of the prism compared to original Original ft 3 SAME x 2 x ,344 ft 3 4,536 ft or 8 times Sorry, I can t give you any more help. 3 3 or 27 times x 4 x 5 4, = What happened?? 27 or 3 3

45 Let s complete our table By George, I Base of think you ve the got of it! the Keep of the triangle triangle going! prism compared to original Original ft 3 SAME x ,344 ft or 8 times x ,536 ft or 27 times x 4 x 5

46 Let s complete our table Go ahead and Base complete of the the triangle last of row. the triangle of the prism compared to original Original ft 3 SAME x ,344 ft or 8 times x ,536 ft or 27 times x ,752 ft or 64 times x 5

47 Let s complete our table Original Did it work? Base of Can we say the that of this the of the triangle pattern triangle is true prism 6 for all prisms? ft 3 compared to original SAME x ,344 ft or 8 times x ,536 ft or 27 times x ,752 ft or 64 times x ,000 ft or 125 times

48 True or False? 1. The volume of a triangular prism is 100m 3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 800m 3. or True False

49 True or False? 2. The volume of a triangular prism is 310 ft 3. If the dimensions are multiplied by a scale factor of 2, the new volume will be 620 ft 3. or True False

50 True or False? 3. The volume of a rectangular prism is 10 km 3. If the dimensions are multiplied by a scale factor of 4, the new volume will be 270 km 3. or True False

51 True or False? 4. The volume of a rectangular prism is 21 yd 3. If the dimensions are multiplied by a scale factor of 3, the new volume will be 567 yd 3. or True False

52 True or False? 5. The volume of a rectangular prism is 50 m 3. If the dimensions are multiplied by a scale factor of x, the new volume will be 50x 3 m 3. or True False

53 On your own 1,080 ft 3 54 cm Suppose the volume of a rectangular prism is 40 ft 3. Find the new volume, if the dimensions are multiplied by a scale factor of Suppose the dimensions of a triangular prism are multiplied by a scale factor of 2 and the NEW volume is 432 cm 3, what was the original volume? 3. Suppose the original volume of a triangular prism is 120 m 3 and the NEW volume is 3,240 m 3, what scale factor were the dimensions multiplied by?

54 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