Unit 3: 2D and 3D Measurement & Optimizing Measurements ISU

Transcription

1 MPM 1DE NAME: Unit 3: D and 3D Measurement & Optimizing Measurements ISU To complete this independent study, you are required to fill in the appropriate information where necessary, work through the given examples, read the information provided and learn the concepts in each of the units (that is, read and learn concepts from lesson 3-1 of this package, then complete the assigned questions for 3-1) etc. Note: Answers are located in your textbook or given on a handout, where necessary. Each exercise will be handed in for assessment on the due dates below. At the end of the unit there will be a test. DUE DATES (on website): Part 1 D & 3D Terminology 3-1 due: (pp. 1-4 in booklet) 3- due: (pp. 5-7 in booklet) 3-3 due: (pp in booklet) Part Optimizing Measurements In class Investigations 3-4 due: (p. 11 in booklet + handouts) Unit 3 Test: Overall Expectations Measurement and Geometry (M&G) M&G1: Determine, through investigation, the optimal values of various measurements (gr. 9) M&G: Solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures (gr. 9)

2 3-1 -Dimensional Geometry Terminology Look at your Grade 9 text pg. 385 and the Glossary AND use a dictionary! POLYGONS Polygon: a two-dimensional closed figure whose sides are line segments CLASSIFICATION OF POLYGONS 3 sides: 7 sides: 4 sides: 8 sides: 5 sides: 9 sides: 6 sides: 10 sides: TRIANGLES Triangle: polygon with 3 sides and 3 angles CLASSIFICATION OF TRIANGLES By Angles: Right Triangle Obtuse Triangle Acute Triangle By Sides: Equilateral Triangle Isosceles Triangle Scalene Triangle Fill in the blanks below with an appropriate definition and diagram: Right Triangle: triangle with a 90 angle Obtuse Triangle: Acute Triangle: Equilateral Triangle: Isosceles Triangle: Scalene Triangle:

3 QUADRILATERALS Quadrilateral: polygon with 4 sides Fill in the blanks below with an appropriate term, definition or diagram: Trapezoid: : quadrilateral in which opposite sides are parallel Rectangle: : parallelogram in which all sides are equal Square: (1) rhombus in which the angles are () Rectangle in which all the sides are Examples: Ex 1) Calculate the area. 4 cm 6 cm 8 cm * include a diagram* 1 cm A = 1 bh b = 1 cm, h = 6 cm * use an appropriate formula & state givens WITH units* = 1 (1)(6) *substitute with brackets but WITHOUT units* = 36 The area is 36 cm * include a concluding statement WITH units*

4 Ex ) Find the value of the variable if A = 69 cm². 8 cm 15 cm h h ( a + b) A = a = 15 cm, b = 8 cm, A = 69 cm h( ( 15) + ( 8 ) (69) = 138 = h (3) 138 = h 3 6 = h The height is 6 cm. Ex. 3) Find the area of the shaded region. 14 mm 1 mm A = A square A triangle = s - 1 bh s = 14 mm, b = 1 mm, h = 14 mm = (14)² - 1 (1)(14) = = 189 The area of the shaded region is 189 mm Ex 4) Find the perimeter of the following. 8 cm 4 cm 1 P = π d + d 1 = π = π b ( 4) + ( 4) + ( 8) d = 4 cm, b = 8 cm The perimeter is 6.3 cm HOMEWORK: Grade 9 Textbook: p. 43 # 1abce, bcde, 3, 4, 10, 1, 15 (Include diagrams with your answers!)

5 3-3-Dimensional Geometry Terminology Solid: a 3-dimensional figure with a closed surface and a measurable volume Cube Cylinder Polyhedron: A 3-dimensional figure with faces that are polygons There are two types of polyhedrons: Prisms & Pyramids Prism: - polyhedron with two parallel, identical, polygonal faces (bases) - they are distinguished from each other by the shape of their base Triangular Prism Cube Rectangular Prism Pentagonal Prism ( Square Prism) Pyramid: a polyhedron whose base is a polygon and other faces are triangles that meet at a common point Square-based Pyramid Triangular-based Pyramid Hexagonal Pyramid

6 ROUND BODIES Round Body: solid that has at least one curved surface There are three types of round bodies: Cylinders, Cones, & Spheres Cylinder: solid in which the two opposite faces are equal circles and the straight line joining their centres is perpendicular to the base Cone: solid with a circular base and a curved lateral surface that extends from the base to a point called the vertex Sphere: solid in which all the points are the same distance from a fixed, interior point called the centre

7 Examples: Ex. 1) Find the volume of a cylinder if d = 1.6 cm and h = 15.8 cm. V = πr h r = 6.3 cm, h = 15.8 cm = π (6.3) (15.8) 15.8 cm 1969 The volume is approximately 1969 cm³ 1.6 cm Ex. ) A silo has a radius of.35 m and a height of 10.4 m. Find the area of the exposed surface of the silo to the nearest m²..35 m () A total = A 1 + A = πrh + πr = π (.35)(7.89) + π (.35) r =.35 m, h = 7.89 m (1) 10.4 m = 48 18π 151 The area of the exposed surface is approximately 151 m². *height of the cylinder is: = 7.89* HOMEWORK: (Include diagrams with your answers!) Grade 9 Textbook: pg. 441 # 1a, a, 3b, 4a, 5, 6, 7, 9 pg. 447 # 1b,, 3, 5, 6 pg 454 # 1b, a, 4, 5 pg. 459 # 1a, 3, 4, 5 pg 465 # 1a, 3, 4

8 3-3 Applications Ex. 1) A cone fits just inside a cylinder. The volume of the cylinder is 9 45 cm³. What is the total surface area of the cone to the nearest square centimetre? V cylinder = π r h r = 10 cm V = 945 cm 3 (9 45) = π (10) h 945 π (10) = h 30 h h + r = s (30)² + (10)² = s 1000 = s ± s h = 30 cm r = 10 cm 0 cm Since s is a side length s > 0 s A cone = π r + πrs = π (10) + π (10)(31.6) = 416π 1307 The area of the cone is 1307 cm² h = 30 cm s = 31.6 cm Ex) A tennis ball fits just inside a plastic cube whose sides measure 40 mm. Determine the volume of the empty space inside the cube. V empty space = V cube V ball = c πr = (40) π (0) = π c = 40 mm r = 0 mm 40 mm 40 mm 40 mm The volume of the empty space is mm³ HOMEWORK: (Include diagrams with your answers!) Grade 9 Textbook: pg. 441 # 11, 16 pg. 447 # 8, 10 pg. 454 # 14 pg. 466 # 7, 8 Worksheet: Application Questions #1-11

9 Application Questions 1. A cone-shaped paper cup with a radius of 5 cm and a height of 1 cm is completely filled with apple juice. The juice is then poured into a cylinder-shaped glass with a radius of 4 cm. Find the minimum height of the second glass so that all of the juice can be contained without spilling.. A cone-shaped party hat is shown below. Find the area of the exterior surface of the hat. Express your answer in terms of π. 6 cm 4 cm 10 cm 3. A salt shaker filled with salt is cylindrical in shape and has a radius of 5 cm and a height of 0 cm. All of the salt is poured out into a cone-shaped pile with a diameter of 16 cm. Find the height of the pile of salt to the nearest tenth of a centimetre. 4. a) A pile of salt has a conical shape. The base of the pile has a diameter of 10 m and the pile is 3 m high. Calculate the volume of the pile to the closest cubic metre. (Use π ) b) The pile of salt from part (a) is placed in a cylindrical container with a radius of 5 m. The salt is levelled in the cylindrical container. Determine the height of the salt in the cylindrical container. (Use π ) 5. A cylindrical container of sugar whose radius is 3 cm and height is 1 cm is poured into a cone-shaped pile with a radius of 8 cm. Calculate the height of the cone-shaped pile. 6. A spherical ball of ground beef is placed in a press to make a cylindrical hamburger patty with a diameter of 8 cm. If the sphere of beef has a radius of 3 cm, calculate the thickness of the hamburger patty. Assume that the volume of the ground beef remains constant. 7. Parmesan cheese is sold in cylindrical containers with a radius of 4 cm and a height of 0 cm. For safety reasons, the entire container must be covered with plastic wrapping. a) How many square centimetres of plastic wrap are required to cover an entire container of cheese? b) The containers are shipped in cardboard boxes. The containers are all placed standing up in the boxes. If each box contains levels of containers and each level is 3containers wide and 4 containers long, determine the minimal dimensions of each cardboard shipping box.

10 8. An observatory is cylindrical in shape and is topped by a hemisphere. The diameter of the base of the observatory is 0 m and the total height is 50 m. 50 m 0 m a) Determine the height of the hemisphere part of the observatory. b) The exterior of the observatory must be painted. Determine the surface area of the observatory in terms of pi. c) If it costs $0.5/m² to paint the exterior of the observatory, how much will it cost to paint the observatory? 9. An ice cream cone produces Drumsticks. These are cones filled with vanilla ice cream and then topped with a hemisphere of chocolate ice cream. The radius of the top of the cone and of the hemisphere of chocolate ice cream is 3 cm. The height of the cone is 1 cm. 3 cm 1 cm a) Calculate the volume of each flavour of ice cream in a Drumstick. Express your answer in terms of π. b) To make a batch of Drumsticks the manufacturer uses 1 containers of chocolate ice cream. How many containers of vanilla ice cream are needed to make the batch? 10. A spherical ball has a surface area of 100π cm². Calculate the volume of the ball. Express your answer in terms of π. 11. Find the total surface area of a closed hemisphere with a radius of 5 cm. Express your answer in terms of π. ANSWERS: cm 7. a) 603 cm² b) 3 cm 4 cm 40 cm. 60π cm² 8. a) 10 m b) 1000π m² c) $ cm 9. a) Vanilla: 36π cm³ Chocolate: 18π cm³ 4. a) 79 m³ b) 1.0 m b) 4 containers cm 500π 10. cm³ cm π cm²

11 3-4 Optimizing Measurments HOMEWORK: (Include diagrams with your answers!) Grade 9 Textbook: p. 488 # 3, 5, 8 p. 495 # 4, 5, 7 p. 50 # a, 5, 7 p. 508 # 3, 4, 6 p. 513 # 4, 5, 6