Chapter 8: Proportional Reasoning

Transcription

1 Chapter 8: Proportional Reasoning Section 8.1 Chapter 8: Proportional Reasoning Section 8.1: Comparing and Interpreting Rates Terminology: Rate: A comparison of two amounts that are measured in different units. Ex. Keying 240 words/8 minutes. Rates from a Graph can be measured using the slope formula: Rate = y 2 y 1 x 2 x 1 Unit Rate: A rate in which the numerical value of the second term is 1. Ex. Keying 240 words/8 minutes expressed as a unit rate is 30 words/min Comparing Two Rates Expressed in Different Units Ex. Natasha can buy a 12 kg turkey from her local butcher for $ The local supermarket has turkeys advertised in its weekly flyer for $1.49/lb. There is approximately 2.2 lb in 1 kg. Which store has the lower price? Ex. Jack can buy 15 gallons of gasoline for $61.56 from the local gas station. However, the next town over has gasoline priced at $1.06/L. There is approximately 3.8 L in 1 gal. Which gas station has the lowest price?

2 Chapter 8: Proportional Reasoning Section 8.1 Solving a Problem Involving Rates Ex. When making a decision about buying a vehicle, fuel efficiency is often an important factor. The gas tank of Mario s car has a capacity of 55L. The owner s manual claims that the fuel efficiency of Mario s car is 100 km/7.6 L on the highway. Before Mario s first big highway trip, he set his trip meter to 0 km so he could keep track of the total distance he drove. He started with the gas tank full. Each time he stopped to fill up the tank, he recorded the distance he had driven and the amount of gas he purchased. Fill-Up Total Distance Driven (km) Quantity of Gas Purchased (L) (a) On what leg of Mario s trip was his fuel efficiency the best? (b) Did the car achieve the manufacturer s fuel efficiency rating of of the trip? Did it achieve it over all? 100 km 7.6 L on either leg

3 Chapter 8: Proportional Reasoning Section 8.1 Rates on a Graph Below is a graph that shows the how far a person drove over the course of the 65 minutes. (a) Determine the rate the person is moving at over each segment of the graph (b) When was the person moving the fastest? (c) Was the person stopped? If so for how long? (d) What is different about the last segment of graph? What could it mean?

4 Chapter 8: Proportional Reasoning Section 8.2 Section 8.2: Solving Problems Involving Rates Solving a Problem Involving Rates Ex. Jeff lives in a town near the Canada U.S. border. The gas tank of his truck holds about 90 L. He can either buy gas in his town at $1.06 Can/L or travel across the border into the United States to fill up at $2.86 U.S./gal. Which is the better option financially? (Note that the 1 gal = 3.79 L and $1 U.S. = $1.32 Can) Connecting Rates to Contextual Situations Describe a situation in which each unit rate might be used. (a) 0.05 mg/kg (b) 98.5 /L (c) 7.2 Mbps

5 Chapter 8: Proportional Reasoning Section 8.2 Reasoning to Solve a Rate Problem Ex 1. Paula is asked to order snacks for an office meeting of 180 people. She decides to order desert squares, which some in boxes of 24. She estimates that she will need 2.5 squares/person. (a) How many boxes should she buy? (b) If each box costs $12.75, how much will the snack order cost? Ex 2. A reception hall offers rates for hosting events. For a dinner, the hall charges $82 per table and each table at the hall can hold 8 guests. What will the cost be for a dinner that will have 160 guests?

6 Chapter 8: Proportional Reasoning Section 8.2 Solving a Problems that Involves Different Rates Amelia walks briskly, at 6 km/h. When she walks at this rate for 2 h, she burns 454 Cal. Rory walks at a slower rate of 4 km/h, burning 62 Cal in 30 minutes. If Amelia walks for 3 h, how much longer would Rory have to walk in order to burn the same number of Calories?

7 Chapter 8: Proportional Reasoning Section 8.3 Section 8.3: Scale Diagrams Terminology: Scale: The ratio of a measurement on a diagram to the corresponding distance measurement on the shape or object represented by the diagram. Scale Diagram: A drawing in which measurements are proportionally reduced or enlarged from actual measurements; A scale diagram is similar to the original. Scale Factor: A number created from the ratio of any two corresponding measurements of two similar shapes or objects, written as a fraction, a ratio, a decimal, or a percent. Scale Factor = Lenght of a Segment in the Scale Diagram Lenght of the corresponding segment in the Original Diagram Scale Fator (SF) = Scale Original Using Scale Diagrams to Determine the Scale Factor Ex. Determine the scale factor. Determine if each is a reduction or an enlargement. (a) A B E F 30 cm 12 cm D Original Diagram 10 cm C H 4 cm S cale Diagram G (b) A 35 mm E 7 mm B 15 mm C D 3 mm F Scale Diagram Original Diagram

8 Chapter 8: Proportional Reasoning Section 8.3 Determining an Unknown Length using Scale Factor Ex. In each, determine the unknown length. (a) A model airplane is 50cm long. It was made using a scale factor of 1:20. Determine the length of the original airplane. (b) A model boat is 2.5 inches in length. It was created using the ratio 1: 144. Determine the length of the original boat. (c) Sara has a picture that is 11 inches long. She wants to get the picture enlarged to three times its original size. Determine the length of the new picture.

9 Chapter 8: Proportional Reasoning Section 8.3 Drawing Scale Diagrams of Enlargements and Reductions A scale diagram is drawn by applying the given scale factor to the original measurements and redraw each side in the same shape and orientation after recalculating. Reduce by a factor of 2 Enlarge by a factor of 1.5 Enlarge by a factor of 3

10 Chapter 8: Proportional Reasoning Section 8.3 Similar Figures Terminology: Similar Figures: Two figures are considered to be similar figures if they satisfy the following two properties: o Their corresponding angles are equal o Their corresponding sides are proportional (This means that they have all been multiplied by the same scale factors) When two figures satisfy these quantities we say they are similar (~) Ex: A E cm 3.2 cm 5.6 cm 6.4 cm D B 3.0 cm 76 C 2.2 cm H F 6.0 cm cm G In this example Quadrilateral ABCD ~ Quadrilateral EFGH, so we know that: AB EF = BC FG = CD GH = DA HE This allows us to determine an unknown side assuming we know the corresponding side as well as at least one other set of corresponding side.

11 Chapter 8: Proportional Reasoning Section 8.3 Ex1. Given that the two figures below are similar, calculate: (a) The length of EF A 9.2 in 81 D E 2.3 in H 7.6 in B F in G (b) The length of BC C (c) The measure of B (d) The measure of H

12 Chapter 8: Proportional Reasoning Section 8.3 Ex2. Given that the two figures are similar calculate the unknown quantities: (a) The measure of D B F m (b) The measure of F C 4.6 m D 9.9 m 44 E 13.8 m G (c) The length of CB (d) The length of FG

13 Chapter 8: Proportional Reasoning Section 8.5 Section 8.4: Scale Factors and Areas of 2D Shapes Area Formulae of Basic Shapes Shape Formula Symbol Meanings Triangle A Triangle = 1 2 bh b = base h h = height b Rectangle w A Rectangle = lw l = length w = width l r Circle A Circle = πr 2 r = radius (note: r = 1 2 d) Parallelogram A Parallel = bh b = base h = height Trapezoid A Trap = 1 2 h(a + b) b = base length h = height a= top length NOTE: You will need to remember these formulae as they will not be provided.

14 Chapter 8: Proportional Reasoning Section 8.5 Determining the Effect of Scale Factor on Area Given each figure shown, determine how the scale factor will affect the Area of the new figure Shape Area Original Scale Factor Scale Diagram Scale Diagram Area What do you notice about the difference between the original area of the figure and the area of the figure after the scale factor is applied? Is it just the area multiplied by the scale factor? Divide the scale diagram areas by the original areas. What do you notice about the result with respect to the scale factor in each example?

15 Chapter 8: Proportional Reasoning Section 8.5 Scale Factors and 2D Shapes If two two-dimensional shapes are similar and their dimensions are related by a scale factor (k), then the relationship between the area of the similar shape and the area of the original shape can be expressed as: Area Scale Diagram = k 2 Area Original If the area of a similar two-dimensional shape and the area of the original shape are known, then the scale factor (k) can be determined using the formula: k 2 = Area Scale Diagram Area Original Determining Area using Scale Factor Ex. Jasmine is making a kite from a diagram. The diagram has a scale factor of 2: 25. The area of the kite in the diagram is 20 cm 2. How much fabric will she need to create her kite? Ex. Mackenzie is building a model car. The model is made using a scale model 1:50. The area of the model car is 70 in 2. Determine the surface area of the actual car.

16 Chapter 8: Proportional Reasoning Section 8.5 Ex. Jim s laptop has a monitor with the dimensions of 9 in by 12 in. The image on his laptop is projected onto the screen of a smartboard. According to the documentation for the smartboard, its area is in 2. (a) Assuming the two images are similar, determine the scale factor used to create the image on the smartboard. (b) Determine the dimensions of the smartboard. Ex. The dimensions of your smartphone is 5.5 in by 3 in. The image of your smartphone is mirrored onto your T.V. The area of your TV is 1056 in 2 according to your user manual. (a) Assuming the mirrored image is similar, determine the scale factor used to create the mirrored image on your TV. (b) Determine the dimensions of your TV.

17 Chapter 8: Proportional Reasoning Section 8.5 Section 8.5: Scale Factors and 3D Objects Volume and Surface Area Formulae Shape Formula Symbol Meanings Rectangular Prism SA = 2lw + 2wh + 2lh V = lwh Triangular Prism SA = bh + la + lb + lc V = 1 2 lbh Cylinder SA = 2πr 2 + 2πh V = πr 2 h Right Pyramid SA = l 2 + 2ls V = 1 3 l2 h Cone SA = πr 2 + πrs V = 1 3 πr2 h Sphere r SA = 4πr 2 V = 4 3 πr3 NOTE: You should remember these formulae as they may not be provided. Note that the Volume formulas for all prisms can be simplified to V = A base h and pyramids V = 1 3 A baseh.

18 Chapter 8: Proportional Reasoning Section 8.5 Scale Factors and 3D Shapes If two three-dimensional shapes are similar and their dimensions are related by a scale factor (k), then the relationship between the volume of the similar shape and the volume of the original shape can be expressed as: Volume Scale Diagram = k 3 Volume Original If the area of a similar two-dimensional shape and the volume of the original shape are known, then the scale factor (k) can be determined using the formula: k 3 = Volume Scale Diagram Volume Original Note: Surface area works the same as regular Area. Determining the Scale Factor with Surface Area and Volume 1. What is the scale factor of the following pairs of similar spheres? a) Volume of the original is 450 mm³ and its image is mm³. b) Surface area of the original is 248 in² and its image is in².

19 Chapter 8: Proportional Reasoning Section 8.5 c) The volume of a model water tank is L the original can hold a volume of L. d) The surface area of the park is m 2 and the surface area of a model of the park is 3.94 m 2. Using Scale factor to determine Volume and Surface Area Ex.1 Brenda is a potter. She is creating two similar vases, with their dimensions related by a scale factor of 3 4. The larger vase has a volume of 9420 cm3. Determine the volume of the smaller vase.

20 Chapter 8: Proportional Reasoning Section 8.5 Ex2. An oil tanker has a capacity of 32 m 3. A similar oil tank has dimensions that are larger by a scale factor of 3. What is the capacity of the larger tank? Ex3. A cereal box has a surface area of 6200 cm 2. A jumbo sized box is enlarged by What is the surface area of the jumbo sized box.

21 Chapter 8: Proportional Reasoning Section 8.5 Ex4. Below is a model water tank. The original tank was created using the scale factor of 1 cm: 1.5 m. (a) What is the total surface area of the original tank (b) What is the total volume of the original tank (c) Was it necessary to determine the dimensions of the original tank to answer either question?