UNIT 7 THE REST! Date Lesson Text TOPIC Homework May 22 7.1 9.2 May 23 7.2 9.3 May 24 7.3 9.4 Optimization Rectangles Optimization Square-Based Prism Optimization Cylinder WS 7.1 Pg. 495 # 2, 3, 5a, 7 Pg. 502 # 2, 3, 6, 7 Pg. 508 # 1-4 Pg. 513 # 1, 2, 5, 6 May 25 7.4 7.1 Angles in Triangles Pg. 371 # 1-9, 11, 14, 15 May 28 7.5 8.2 May 29 7.6 8.2 May 30 7.7 8.3 Angles in Quadrilaterals QUIZ (7.1 7.3) Angles in Polygons Angles and Algebra QUIZ (7.4 7.6) Pg. 381 # 1-3, 5-7 Pg. 391 # 1-6, 9, 10, 13abce WS 7.7 May 31 7.8 June 1 7.9 CBR/Distance-Time Graphs Pg. 392 # 3, 4, 5, 7, 8 CBR/Distance-Time Graphs WS 7.9 June 4-5 7.10 Unit 7 Review June 6 7.11 UNIT 7 TEST June 7-13 June 14-15 June 18-19 June 22 EQAO & Exam Review EQAO MORE EXAM REVIEW EXAM Room 117 @ 9:30
MPM 1D Lesson 7.1 Optimization of Rectangles OPTIMIZATION the process of finding values that make a given quantity the greatest or least possible given certain conditions How can you model the maximum area of a rectangle with a fixed perimeter? 1. How can you model the areas of rectangles with the same perimeter? You have 12 m of rope to fence off a rectangular play area at a summer camp. Complete the table below for whole number lengths only. Rectangle Width (m) Length (m) Perimeter (m) Area (m 2 ) 1 12 2 12 3 12 4 12 5 12 c) Is it possible to construct other rectangles? d) Which rectangle had the greatest area? How can you model the maximum area of a rectangle with a fixed sum of the lengths of three sides? Brandon's customer decides to use an existing hedge as one of the sides of the enclosure. This means that there will be fencing on three sides. They still want the area to be as large as possible. Do you think he will be able to enclose more, less or the same amount of area?
Complete the table. Rectangle Width (m) Length (m) Sum of Lengths of Three Sides (m) Area (m 2 ) 1 1 10 12 30 2 2 8 12 3 3 6 4 4 4 5 5 2 What are the dimensions of the rectangle with maximum, or optimal area? KEY POINTS Optimizing the area of a rectangle means finding the dimensions of the rectangle with maximum area for a given perimeter. For a rectangle with a given perimeter, the dimensions that result in a maximum area are equal ie: a square The dimensions of a rectangle with optimal area depend on the number of sides to be fenced. If fencing is not required on all sides, a greater area can be fenced. If three sides are to be fenced the dimensions that result in a maximum area are a rectangle whose length is twice the width. For a given perimeter, the shape that maximizes the area is a SQUARE. or as close to a square as possible. P 4s, where s is the side length For a given perimeter of a 3-sided rectangle, the shape that maximizes the area is a RECTANGLE WHOSE LENGTH IS TWICE THE WIDTH. P 4w, where w is the width of the rectangle
How can you model the Minimum Perimeter for a Given Area? A rectangular dog exercise area is to have an area of 36 m 2. Complete the table below for whole number lengths only. Rectangle Width (m) Length (m) Perimeter (m) Area (m 2 ) 1 1 36 2 2 36 3 3 36 4 4 36 5 6 36 What dimensions use the least amount of fencing? For a given area, the shape that minimizes the perimeter is a SQUARE. A s 2, where s is the side length Ex. 1 What is the maximum area of a rectangle with perimeter 30 m? Solution The maximum area occurs when the rectangle is a square. Determine the side length, s, of a square with perimeter 30 m.
Ex. 2 What is the minimum perimeter for a rectangle with area 40 m 2? State your answer to 1 decimal place. Solution The minimum perimeter occurs when the rectangle is a square. Determine the side length, s, of a square with area 40 m 2. Ex. 3 A rectangular swimming area is to be enclosed by 100 m of rope. One side of the swimming area is Along the shore, so the rope will only be used on three sides. What is the maximum swimming area that can be created? Solution The maximum area occurs when the rectangle has a length that is twice the width. Determine the width, w of a rectangle with perimeter of 100 m. WS 7.1
MPM 1D Lesson 7.2 Optimizing a Square Based Prism Investigation How can you compare the surface areas of square based prisms with the same volume? A square based prism has a volume of 64 cubic units. Complete the table below. Prism Length Width Height V h l w Volume Surface Area 1 1 1 64 64 2 2 2 64 3 3 3 64 4 4 4 64 5 5 5 64 6 6 6 64 7 7 7 64 8 8 8 64 SA 2( lh lw wh) Which square based prism has the minimum, or optimal surface area? Describe the shape of this prism. Predict the dimensions of the square based prism with minimum surface area if you have a volume of: a) 8 cm 3 b) 27 cm 3 c) 125 cm 3 d) 250 cm 3 (correct to 2 decimal places) KEY POINTS Minimizing surface area is important when designing containers to save on material and reduce heat loss. For a square based prism with a given volume, the minimum surface area occurs when the prism is a cube. Given a volume, you can find the dimensions of the square based prism with the minimum surface area by solving for s in the formula V = s 3, where V is the given volume and s is the side length of the cube.
Ex. 1 a) A popcorn company ships popcorn in large cardboard boxes with a volume of 500 000 cm 3. Determine the dimensions of the box that will use the least cardboard. b) Find the amount of cardboard necessary to make the box, correct to the nearest tenth of a square metre. Describe any assumption you have made. Ex. 2 To keep heat loss to a minimum, hot food is transported in containers that have the least surface area. That way there is less area for the heat to escape the container. a) Find the dimensions of the square based container with a volume of 165 000 cm 3 that will minimize heat loss. b) What other factors might be considered when designing the container?
Maximizing the Volume of a Square Based Prism Prism Side Length of Base (cm) Area of Base (cm 2 ) Surface Area (cm 2 ) Height (cm) Volume (cm 3 ) V = l x w x h 1 24 2 24 3 24 Which square based prism has the maximum volume? Describe the shape of this prism. Prism Side Length of Base (cm) Area of Base (cm 2 ) Surface Area (cm 2 ) Height (cm) Volume (cm 3 ) V = l x w x h 1 54 2 54 3 54 4 54 5 54 What conclusion can you make about the maximum volume of a square based prism with a given surface area?
Ex. a) Determine the dimensions of a square based prism with a maximum volume that can be formed using 5400 cm 2 of cardboard. b) What is the volume of the prism? Ex. Determine the maximum volume of the square-based prism that can be made using 5400 cm 2 of cardboard. KEY POINTS For a square based prism with a given surface area, the maximum volume occurs when the prism is a cube. The surface area of a cube is given by SA = 6s 2, where s is the side length of the cube. When you are given the surface area, solve for s to find the dimensions of the square based prism with the maximum volume. Pg. 495 # 2, 3, 5a, 7 & Pg. 502 # 2, 3, 6, 7
MPM 1D Lesson 7.3 Optimizing a Cylinder Investigate How can you compare the volumes of cylinders with the same surface area? 1. Design a cylindrical can that uses no more than 375 cm 2 of aluminum that has the greatest capacity possible Start with the formula for the surface area of a cylinder SA = 2πr 2 + 2πrh Rearrange the formula to solve for h. The steps have been done for you. Write a short explanation of each step. The first has been done for you. SA = 2πr 2 + 2πrh STEPS 375 = 2πr 2 + 2πrh Substitute 375 for SA 375 2πr 2 = 2πr 2 + 2πrh 2πr 2 375 2πr 2 = 2πrh 2 375-2πr 2πrh = 2πr 2πr 2 375-2πr h= 2πr 2. Complete the table below using the formula for volume of a cylinder V = πr 2 h. Cylinder Radius (cm) Height (cm) Volume (cm 3 ) Surface Area (cm 2 ) 1 1 375 2 2 375 3 3 375 4 4 375 5 5 375 6 6 375 7 7 375 3. a) What is the maximum volume for the cans in your table? b) What are the radius and height of the can for this volume? r = h =
KEY POINTS The maximum volume for a given surface area of a cylinder occurs when its height equals its diameter. That is h = d or h = 2r. The dimensions of the cylinder with maximum volume for a given surface area can be found by solving the formula SA = 2πr 2 + 2πrh = 2πr 2 + 2πr(2r) = 2πr 2 + 4πr 2 SA = 6πr 2 for r, and the height will be 2r. Ex. a) Determine the dimensions of the cylinder with maximum volume that can be made with 600 cm 2 of aluminum. State your answer correct to the nearest hundredth of a cm. b) What is the volume of the cylinder to the nearest cubic cm.
Investigate How can you compare the surface area of cylinders with the same surface volume? Your task is to construct a cylinder with a volume of 500 cm 3 which has the least surface area. Complete the table below using the formula V cylinder = (area of base)(height). Area of base = V h Cylinder Radius (cm) Base Area (cm 2 ) Volume (cm 3 ) Height (cm) Surface Area (cm 2 ) 1 2.5 500 2 4.3 500 3 6.1 500 Which cylinder has the least surface area? How does its height compare to its diameter? KEY POINTS The minimum surface area for a given volume of a cylinder occurs when its height equals its diameter. That is h = d or h = 2r. The dimensions of the cylinder minimum surface area for a given volume can be found by solving the formula V = πr 2 h, h = 2r = πr 2 (2r) V = 2πr 3 for r, and the height will be 2r. Ex. a) Determine the least amount of aluminum required to construct a cylindrical can with a 1 L capacity, to the nearest square centimetre. b) Describe any assumptions you made. Pg. 508 # 1-4 & Pg. 513 # 1, 2, 5, 6
MPM 1D Lesson 7.4 Angles in Triangles The 3 interior angles of a triangle have a sum of 180 Classifying Triangles -by side length Scalene Triangle A triangle where all sides have a different length. Isosceles Triangle A triangle where two sides have the same length. Equilateral Triangle A triangle where all sides have the same length all angles are 60. -by angle measure Acute Triangle A triangle in which all angles are less than 90. Obtuse Triangle A triangle which contains one obtuse angle. an angle between 90 and 180 Right Triangle A triangle in which one angle is 90. Ex.1 Find the measure of each indicated exterior angle. a) b) c)
d) e) f) g) h) i) Ex. 2 Find the measure of each indicated exterior angle. a) b)
c) d) Ex. 3 Find the measure of each indicated angle. a) b) c) d) Ex. 4 One interior angle in an isosceles triangle measures 42. Find the possible measures for the exterior angles. Pg. 371 # 1-9, 11, 14, 15
MPM1D Lesson 7.5 Angles in Quadrilaterals Any quadrilateral can be divided into 2 triangles. The sum of the angles in each triangle is 180. The sum of the angles in 2 triangles is: 2 x 180 = 360 Vertically Opposite Angles are the angles opposite each other when two lines intersect (cross). "Vertical" in this case means they share the same Vertex (or corner point), not the usual meaning of up-down. Vertically opposite angles are equal. In the diagram to the right, 1 = 3 and 2 = 4 When a transversal intersects two lines, four pairs of opposite angles are formed. The angles in each pair are equal. When the lines are parallel, angles in other pairs are also equal. We can use tracing paper to show these relationships. Corresponding angles are equal. They have the same position with respect to the transversal and the parallel lines.
Alternate angles are equal. They are between the parallel lines on opposite sides of the transversal. Interior angles have a sum of 180. They are between the parallel lines on the same side of the transversal. We can use these relationships to determine the measures of other angles when one angle measure is known. Two Angles are Supplementary if they add up to 180 Together they make a straight angle. Angles on one side of a straight line will always add to 180. Two Angles are Complementary if they add up to 90 (a Right Angle).
Ex. 1 Fill in the measures of all interior and exterior angles: Ex. 2 Find the measure of each indicated angle: a) b) Ex. 3 Find the measure of each indicated exterior angle. a) b)
c) d) Ex. 4 Find the measure of each indicated angle. a) b) c) d)
Ex. 5 Find the measure of each indicated angle. a) b) c) d) Pg. 381 # 1-3, 5-7
MPM1D Lesson 7.6 Angles in a Polygon Regular Polygon -All sides of equal length # of sides # of Triangles Sum of Interior Angles Measure of each angle Sum of Exterior Angles
Ex. In a regular dodecagon, 12 sides, what will be the measure of each interior angle? Pg. 391 # 1-6, 9, 10, 13abce
MPM1D Lesson 7.7 Angles and Algebra Ex. 1 Solve for x and then determine the measure of every interior angle. a) b) Ex. 2 Solve for x and then determine the value of m, n, p, and q
Ex. 3 Solve for each variable. a) b) c) d)
e) f) WS 7.7
MPM1D Lesson 7.8 Distance Time Graphs The bell goes and Homer stops work and runs 250m to his car, this takes him 5 minutes. Homer drives for 10 minutes but only gets 2000m before he needs to stop for a doughnut. He sits in his parked car and spends 5 minutes eating the doughnut. Finally he starts the engine and drives the 1000m back home in 5 minutes. Distance travelled (m) 3500 3000 2500 2000 1500 1000 500 0 0 5 10 15 20 25 Time taken (mins) Lisa sees the time and spends 5 minutes packing her saxophone away. It s then a 5 minute walk to her bike 300m away. She then cycles on a flat 800 metres in 5 minutes. At this point she reaches a hill and slows down, travelling the next 300 metres in 3 minutes, before reaching home. Distance travelled (m) 1600 1400 1200 1000 800 600 400 200 0 0 2 4 6 8 10 12 14 16 18 20 Time taken (mins)
Distance from house (m) Maggie crawls out the house and crawls 500m away from the house, this takes 20 minutes. She then sneaks into a lady s handbag for 5 minutes and travels 500m further away from her house. When she gets out, her homing instinct makes her crawl 200m back towards her house, this takes 5 minutes. Maggie is tired so she stops and sucks her soother for 10 minutes. She then starts crawling back covering 200m in 10 minutes, she continues at this speed until she arrives back home. 1000 800 600 400 200 0 0 10 20 30 40 50 Time taken (mins) 60 70 80 Distance-Time Graphs (fill in following graphs together) d d d d d t t t t t Not moving Moving forward Moving backwards Moving forward Moving forward quickly slowly
Describe the movement in each section of the graph. Pg. 392 # 3, 4, 5, 7, 8