[PDF] - Slide 7 / 108 Slide 8 / 108 Does a prism need to be a right prism PDF Document

SLIDE 1

Slide 1 / 108

This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.

Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

Slide 2 / 108

Volume

www.njctl.org March 7, 2012

Slide 3 / 108

Slide 4 / 108

Volume of a Prism

Return to Table of Contents

Slide 5 / 108

Definition of Volume -

The amount of cubic units that a solid can hold.

Where area used square units, volume will use cubic units.

Slide 6 / 108

Base height Base

L w h

V = Bh Specific Prisms Box: V = Lwh Cube: V=s3

Finding the Volume of a Prism

SLIDE 2

Slide 7 / 108

Does a prism need to be a right prism for the volume formula to work?

Think of a ream of paper Stacked nicely it has 500 sheets. If the stack is fanned, it still has 500 sheets. So the volume doesn't change if the prism, stack of paper, is right or oblique. The formula V=BH works for all prisms

Slide 8 / 108

Example: Find the volume of the box with length 2, width 6, and height 5.

Slide 9 / 108

Example: The volume of a box is 48 ft3. If the height is 4ft and width is 6ft, what is the length.

Slide 10 / 108

Example: Find the volume of the cube with edges

f 7.

Slide 11 / 108

Example: Find the volume of a cube is 64 m3, what is area of one face?

Slide 12 / 108

Example: Find the volume of the prism with height 8 and hexagon base with apothem 4.

SLIDE 3

Slide 13 / 108

1 What is the volume of a box with edges of 4, 5, and 7?

Slide 14 / 108

2 What is the volume of a box with edges of 8, 6, and 10?

Slide 15 / 108

3 What is the volume of a cube with edges of 5 units?

Slide 16 / 108

4 If the volume of a box is 64 u3 and has height 8 and width 4, what is the length?

Slide 17 / 108

5 If the volume of a box is 120 u3 and has height 6 and length 4, what is the width?

Slide 18 / 108

6 If a cube has volume 27 u3, what is the cubes surface area?

SLIDE 4

Slide 19 / 108

7 Find the volume of the prism.

15 12 20

Slide 20 / 108

8 Find the volume of the prism.

7 2 6 6 6

Slide 21 / 108

Volume of a Cylinder

Return to Table of Contents

Slide 22 / 108 Finding the Volume of a Cylinder

base base height

r r

V = Bh V = πr2h

Slide 23 / 108

Example: Find the volume of the cylinder with radius 4 and height 11.

Slide 24 / 108

Example: The surface area of a cylinder is 425 u2, what is the volume?

SLIDE 5

Slide 25 / 108

9 Find the volume of the cylinder with radius 6 and height 8.

Slide 26 / 108

10 Find the volume of the cylinder with circumference 18π units and height 6?

Slide 27 / 108

11 Find the volume of the cylinder with a surface area 108 u2.

Slide 28 / 108

12 The volume of a cylinder is 108 u3, what is the surface area?

Slide 29 / 108

13 The height of a cylinder doubles, what happens to the volume?

A

Doubles

B

Quadruples

C

Depends on the cylinder

D

Cannot be determined

Slide 30 / 108

14 The radius of a cylinder doubles, what happens to the volume?

A

Doubles

B

Quadruples

C

Depends on the cylinder

D

Cannot be determined

SLIDE 6

Slide 31 / 108

15 A 3" hole is drilled through a solid cylinder with a diameter of 4" forming a tube. What is the volume of the tube?

24" 4" 3"

Slide 32 / 108

Volume of a Pyramid

Return to Table of Contents

Slide 33 / 108 Finding the Volume of a Pyramid

Square Base (B) Slant Height (l ) Pyramid's Height (h)

V = 1/3 Bh

Slide 34 / 108

Example: Find the volume of the pyramid. 5 4 6

Slide 35 / 108

Example: Find the volume of the pyramid. 5 4 6

Slide 36 / 108

Example: Find the volume of the pyramid. 8 8 5

SLIDE 7

Slide 37 / 108

16 Find the volume of the pyramid.

7 6 5

Slide 38 / 108

17 Find the volume of the pyramid.

6 6 8

Slide 39 / 108

18 Find the volume of the pyramid.

12 12 10

Slide 40 / 108

A truncated pyramid is a pyramid with its top cutoff parallel to its base. Find the volume of the truncated pyramid shown. 2 2 6 6 9 3

Slide 41 / 108

19 Find the volume of the pyramid.

2 2 8 8 12 3

Slide 42 / 108

Volume of a Cone

Return to Table of Contents

SLIDE 8

Slide 43 / 108

Finding the Volume of a Cone

r height S l a n t H e i g h t

l

V = 1/3 Bh V = 1/3 π r2 h

Slide 44 / 108

Example: Find the volume of the cone. 9 r= 7

Slide 45 / 108

Example: Find the volume of the cone. 12 r= 4

Slide 46 / 108

Example: Find the volume of the cone, with lateral area 15π units2 and slant height 5.

Slide 47 / 108

20 What is the volume of the cone?

8 d=10

Slide 48 / 108

21 What is the volume of the cone?

10 40o

SLIDE 9

Slide 49 / 108

22 What is the volume of the truncated cone?

r=8 r=4 6 6

Slide 50 / 108

Volume of a Sphere

Return to Table of Contents

Slide 51 / 108 Finding the Volume of a Sphere

r V = 4/3 π r3

Slide 52 / 108

Example: Find the volume of the sphere.

9

Slide 53 / 108

Example: Find the volume of the sphere. Great Circle: A=25π u2

Slide 54 / 108

Example: Find the volume of the sphere. Surface Area: SA=36π u2

SLIDE 10

Slide 55 / 108

23 Find the volume of the sphere.

4

Slide 56 / 108

24 Find the volume of the sphere.

6

Slide 57 / 108

25 Find the volume of the sphere.

Great Circle: A= 16π u2

Slide 58 / 108

26 Find the volume of the sphere.

Surface Area: SA= 16π u2

Slide 59 / 108

27 Find the volume of the sphere.

Cross Section: A= 16π u2 3

Slide 60 / 108

Cavalieri's Principle

Return to Table of Contents

SLIDE 11

Slide 61 / 108 Cavalieri's Principle

If two solids are the same height, and the area of their cross sections are equal, then the two solids will have the same volume

Slide 62 / 108

14 14 14 Which solid has the greatest volume?

Slide 63 / 108

A sphere is submerged in a cylinder. What is the volume of the cylinder not occupied by the sphere? volume of cylinder - volume of sphere The result shows that the left over volume is equal to 2 of what other solid? According to Cavalieri, what can be said about the cross section?

Slide 64 / 108

28 These 2 surfaces have the same volume, find x.

11 11

Slide 65 / 108

29 These 2 surfaces have the same volume, find x.

12 12

Slide 66 / 108

Corresponding Parts of Similar Solids

Return to Table of Contents

SLIDE 12

Slide 67 / 108

Corresponding parts of similar figures are similar. The prisms shown are similar. Find x and y. 4 6 x 9 y 2 The ratio of similitude, k, is the common value that is multiplied to preimage to get to the image. If the smaller prism is the preimage, what is k? If the larger prism is the preimage, what is k?

Slide 68 / 108

30 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of x.

8 8 16 h 2 x y 3

Slide 69 / 108

31 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of y.

8 8 16 h 2 x y 3

Slide 70 / 108

32 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of h.

8 8 16 h 2 x y 3

Slide 71 / 108

4 6 6 9 3 2 Consider the example of the prisms from earlier. The ratio of similitude from the smaller surface to the larger is 3/2. Look at the area of their bases: how do they compare? How do their volumes compare?

Slide 72 / 108

Comparing Similar Figures length in image length in preimage

= k

area in image area in preimage

= k2

volume in image volume in preimage

= k3

SLIDE 13

Slide 73 / 108

Example: r = 9 r = 3 How many times bigger is the great circle on the right? How many times bigger is the surface area on the right? How many times bigger is the voklume on the right?

Slide 74 / 108

33 The scale factor of 2 similar pyramids is 4. If the area of the base of the larger one is 64 u2, what is area of the smaller one?

Slide 75 / 108

34 The scale factor of 2 similar right square pyramids is 3. If the area of the base of the larger one is 36 u2 and its height is 12, what is lateral area of the smaller one?

Slide 76 / 108

35 An architect builds a scale model of a home using 2 in to 5 ft. scale. Given the view of the roof of the model, how much roofing material is needed for the house?

12in 6in 8in 5in 4in 3in

Slide 77 / 108

Coordinates in Space

Graphing Distance Midpoint Diagonal of a Box Equation of a Sphere

Return to Table of Contents

Slide 78 / 108

Graphing in space requires the x-, y-, and z-axes. z +

y

+

x

+

SLIDE 14

Slide 79 / 108

To graph an ordered triple, (x, y, z), draw a box with a vertice at the origin. z +

y

+

x

+

Graph (2, 4, 3)

(2,0,0) (2,4,0) (0,4,0) (0,4,3) (0,0,3) (2,0,3) (2,4,3)

Slide 80 / 108

To graph an ordered triple, (x, y, z), draw a box with a vertice at the origin. z +

y

+

x

+

Graph (-3, -1, -4)

(-3,0,0) (0,-1,0) (-3,-1,0) (-3,0,-4) (0,0,-4) (0,-1,-4) (-3,-1,-4)

Slide 81 / 108

z +

y

+

x

+

Graph (2, 4, 9)

Slide 82 / 108

z +

y

+

x

+

Graph (-1, -4, 0)

Slide 83 / 108

36 What is the ordered triple that was graphed?

A

(2,3,4)

B

(3,2,4)

C

(4,3,2)

D

(3,4,2)

z +

y

+

x

+

4

2 3

Slide 84 / 108

37 What is the ordered triple that was graphed?

A

(-2,3,4)

B

(3,-2,4)

C

(4,3,-2)

D

(3,4,-2)

z +

y

+

x

+

2

4 3

SLIDE 15

Slide 85 / 108

Distance Formula

The distance between two points in space:

Slide 86 / 108

Example: Find the distance between (4,1,-5) and (-2, 8,-2)

Slide 87 / 108

38 What is the distance between (4,-2,5) and (3,5,-6)?

Slide 88 / 108

39 What is the distance between (-1,-2,-3) and (5,0,-4)?

Slide 89 / 108

Midpoint Segment

The midpoint of a segment is found by

Slide 90 / 108

Example: Find the midpoint of (4,3,-8) and (-6,0,9)

SLIDE 16

Slide 91 / 108

40 What is the midpoint of (4,8,10) and (6,4,-12)?

A

(5,6,-1)

B

(10,12,-2)

C

(2,4,22)

D

(1,2,11)

Slide 92 / 108

41 What is the midpoint of (6,-1,5) and (6,5,-1)?

A

(0,-6,4)

B

(0,-3,2)

C

(12,4,4)

D

(6,2,2)

Slide 93 / 108

Diagonal of a Box

h

w

l

Slide 94 / 108

Example: Find the diagonal of the box.

6

4

7

Slide 95 / 108

42 Find the length of the diagonal.

8 5 1 Slide 96 / 108

43 Find the length of the diagonal.

6 4 9

SLIDE 17

Slide 97 / 108

Why is there no slope formula for 3 dimensional geometry? If a line went through the point and had a slope of 4 what would that mean? z +

y

+

x

+

Slide 98 / 108

Spheres

Recall the definition of a circle: Circle- the set of points in a plane a given distance from a given point. The given distance was the radius and the given point was the center. A sphere has a similar definition: Sphere- the set of points a given distance from a given point. The difference is a circle is a plane figure, a sphere is a space figure.

Slide 99 / 108

The Equation of a Sphere

The equation of a sphere is to that similar that of a circle. Where (h,k,j) is the center and r is the radius. { (x,y,z) represent all of the ordered triples that lie on the sphere.}

Slide 100 / 108

Given the equation (x-3)2 +(y+4)2+ z2 = 49 What is the center? How long is the radius? Is the point (4,2,2) inside, on, or outside the sphere? (,3,-4,0) 7 (4-3)2 +(2+4)2+ (2)2 ? 49 (1)2 +(6)2+ (2)2 ? 49 1+36+4 ? 49 41<49 so inside Name a point on the sphere. ex: (3,-4,7), (3,3,0), or (-4,-4,0)

Slide 101 / 108

44 What is the center of

A B C D

Slide 102 / 108

45 What is the radius of