NOTES AND EXERCISES WEEK 9 AND 10
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1 ESSENTIAL MATHEMATICS 3 NOTES AND EXERCISES WEEK 9 AND 10 Scale Drawings A scale drawing is usually a reduction of a real object, such as a building, but can be an enlargement of a very small object, such as a computer chip. Scales on a diagram are usually given as; A statement such as 1 cm represents 2 m A ratio such as 1:100 Example Kel is using a map with the scale 1 cm representing 2 km. How far would he have to walk if the distance on the map is 6 cm? We need to multiply the scaled distance by the scale. 6 2 km = 12 km. Exercise 1 Q1. By measurement and calculation, find the real length of these objects. a) b) c) d) e)
2 Q2. A house is drawn to scale as shown. A house is drawn to scale as shown. (a) What does the scale of 1: 100 mean? (b) If the height of the door in the drawing is 18 mm, what is its actual height? (c) Find the actual width of: (i) the house (ii) a window (d) Find the actual height above ground level of: (i) the ceiling (ii) the peak of the roof
3 Q3. When the ratio for the scale is given in the reverse order, this means that the object is smaller than the scale drawing. In this case we divide by the scale. This drawing of a screw is drawn to a scale 5:1. Scaled length of the screw = cm Actual length of the screw = 5 = cm Q1. By measurement and calculation, find the real length of these objects. a) b) c) d) e) f) Bacteria 100:1 Snowflake 25:1 Floor Plans
4 Plans for buildings are one of the most common uses of scale diagrams. Each house, office block, school or any other building must have plans drawn up before it can be built. House plans use many different symbols and anbbreviations. They are either drawn to scale or have measurements written on them. Measurements on building plans are usually shown in millimetres to avoid the use of decimal points as decimal points can lead to errors in printing and reading. Exercise 2 Q1. Use this floorplan to answer questions in parts a to k. a) What does each of these features on the plan represent? b) Three areas on the plan are covered in small squares. What do the areas have in common? c) How many bedrooms are in the house? d) Which bedrooms have built-in wardrobes? e) Where is the laundry?
5 f) How many doorways lead into the bathroom? g) Which room is 5 29 m long and 3 29 m wide? h) What are the dimensions of the garage, in metres? i) Can a person standing at the front door see into the bathroom? j) Which rooms will get sun in the morning? k) Which rooms will get no direct sunlight? Q2. Use this house plan to answer the questions in parts a to j. a) How many doors are shown on the plan? b) What are the length and width of the house? c) Calculate the width of the kitchen.
6 d) Calculate the floor area of the house. e) What is the ratio of the area of bedroom 1 to the area of bedroom 2? f) What percentage of the floor area of the house is the area of bedroom 1? g) The builder is going to put tiles on the family room floor. What is the area of the family room floor? (i) The builder always buys 10% more than the floor area when he buys tiles. How many square metres of tiles will he buy for the family room? h) The roof guttering, which goes all the way around the house, will cost $24.75 per metre. Calculate the cost of the guttering i) The floor area of the house is 25% of the area of the block of land. What fraction is the same as 25%? (ii) How many times bigger than the area of the house is the area of the block of land? (iii) Calculate the area of the block of land. j) The block of land is a rectangle. What could the dimensions of the block be?
7 Q3. Match each word in the left column to its correct meaning in the right column. Producing Scale Drawings When we want to make scale drawings, we usually start with a rough sketch that has the measurements we need on it. We then choose a scale and draw the objects accurately. We can also use our scale drawings to find other measurements. Example 1 This is rough sketch of Farmer Fred s field. Draw a scale drawing of Farmer Fred s field. A suitable scale to use is 1 cm = 200 m We can use this to calculate the scaled length for each measurement by dividing by the scale = 4.5 cm = 3.5 cm = 6 cm = 1.75 cm Use these measurements to draw the scale drawing.
8 Example 2 Sue is going on a bushwalk from her camping site. She walks 2.5 km due East then 1.9 km due Northeast. a) Draw a scale diagram of Sue s walk b) By measurement calculate how far Sue is from her camp site. A suitable scale is 1 cm = 0.5 km = 5 cm = 3.8 cm The angle between East and Northeast is 45. Construct a diagram using a ruler and protractor. Complete the triangle and measure the third side. This turns out to be 3.5 cm = 1.75 thus Sue is 1.75 km from her camp site. Exercise 3 Answer the following questions on blank paper. Q1. This diagram is not to scale. Make a scale drawing of this field using 1 cm = 500 m. Q2. Keith and Rob are setting off on a hike. They walk 5 km due West of their starting point and then turn and walk 7 km South. They stop for lunch and then walk another 6 km in a North-easterly direction before stopping for afternoon tea. a) Make a scale diagram of their walk using a scale of 1 cm = 1 km. b) How far are they from their starting point?
9 Q3. Construct a scale diagram to calculate the actual height of the hot air balloon above the ground. Use a scale of 1 cm = 50 m. Q4. An old woden gate needs a giagonal brace for support. Construct a scale drawing of the gate and find the actual length of the brace. Use a scale of 1 cm = 40 cm.
10 Q5. Michael wants to swim across the river. Draw a diagram and calculate the width of the river. Use a scale of 1 cm = 10 m. Q6. A golf course has a large lake as an obstacle. Most golfers follow the dog leg around the lake. How far is it in a straight line across the lake, from the tee to the hole?
11 Interior Decorating Good interior decorating can make a huge difference to how a property looks and feels. Colour schemes, paintwork, carpets blinds, curtains and accessories all contribute to the overall look. Border Designs Many decorators place a patterned strip around the top of walls or tiles. This patterned strip is called a frieze and it is usually placed at the top of walls, although it can be placed at any height. A frieze pattern can be made from wallpaper or tiles. Exercise 4 Q1. The diagram shows the frieze Cindy plans to make with black and white tiles for her bathroom. a) How many tiles long is the pattern repeat. b) What colour will each of the tiles labelled i through viii be? c) How many black tiles and white tiles will Cindy need to make a frieze 24 pattern repeats long? d) What fraction of the tiles Cindy will use are black? Q2. This piece of a frieze pattern was found in an old Egyptian building.
12 a) Complete six more tiles on each side of the piece shown. b) What fraction of the tiles the Egyptian builders used to make this frieze were dark (actually purple)? c) Each pattern repeat is 48 cm wide. Calculate the size of each tile. Q3. The walls of a Maths classroom are being decorated with this frieze pattern made from octagons and equal-sized right-angled triangles. a) How wide are the octagons used in the pattern? b) When Aaron decorated his section of the classroom with the frieze he used 36 octagons. How many right-angled triangles did he use? c) How long was Aaron s frieze? d) What is the height of the frieze?
13 Q4. The photograph shows the frieze in a baby s nursery. The pattern repeat in the frieze is 50 cm long. a) Counting the two bears with arms around each other as one bear picture, how many bear pictures are in one pattern repeat? b) If the dimensions of the room are 4.2 m times 3.8 m, how many patterns would be needed to go around the room? Q5. Outdoor paving can be laid in many patterns. Four of these are shown below. Complete the pattern in each grid.
14 Trigonometry The ancient Greeks, between the years 600 BC and 200 AD, laid the foundations of a new branch of mathematics, one that uses angles, triangles and circles to calculate lengths and distances that cannot be measured physically (as they are to large). This new mathematics is now called trigonometry, from the Greek words trigon and metron, meaning triangle and measure respectively. Trigonometry is used widely today to calculate lengths and angles -in engineering, surveying, navigation, astronomy, electronics and construction. Pythagoras Theorem The Greek mathematician Pythagoras ( BC) is credited with discovering the following rule about the sides of a right-angled triangle: c 2 = a 2 + b 2 (hypotenuse) 2 = (side) 2 + (other side) 2 The square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle. Labelling Triangles Note that in the diagram above, the angles of the triangle are labelled by capital letters A, B,C, while the sides are labelled by lower case letters a, b, c. Also, the side with the lower case letter (e.g. a) is always opposite the angle with the corresponding capital letter (e.g. A). This is a convention (accepted rule or agreement) of triangle geometry.
15 Example 1 Determine whether these triangles are right-angled. a) b) The longest side is 8 cm. The longest side is 10 cm. Pythagoras theorem works. Thus this triangle is right- Pythagoras theorem does not work. Thus this triangle is not right-angled. angled. Exercise 5 Q1. Write Pythagoras theorem for these right-angled triangles. a) b) c)
16 Q2. Determine whether each triangle is right-angled. a) b) c) d) e)
17 2018 EM3 Week 9 Investigation 1. In the scale drawing above, what is the dimension of each square? 2. What is the height of the man? 3. What is the height of the sauropod? 4. Measure the height of the sauropod in cm. 5. Use these measurements to calculate the scale of this drawing. Show working. 6. The man s height is what percentage of the sauropods height? Show working.
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