Topic 8: Measurement

137 Topic 8: Measurement Topic 1 Integers Topic 2 Decimals Topic 3 Fractions Topic 4 Ratios Topic 5 Percentages Topic 6 Algebra Topic 7 Equations and Formulae Topic 8 Measurement Duration 2 weeks Content outline PART 1 Introduction Base Units in the Metric System Prefixes Converting Units Rates PART 2 Scientific Notation Perimeter Area Volume TEP023 Foundation Mathematics

138 Topic 8: Measurement Introduction People used to measure their world against the most convenient tool, their bodies. Things were measured in arm or foot lengths. This does make problems if you think you are getting a 15 hand sheep but my hands are small, so it only measures 13 of your hands. Imagine the argument. People in different places used different measurement systems, such as pounds and yards. The problem with this system of course is that it is difficult to compare things in different places. This makes selling things overseas very complicated. So in 1790, the French Academy of Sciences was asked to make a system that everyone could use across the world. The new system was based on the number 10 to make it easy to work with. The official standard for the metre to be used internationally is held in France and is kept in Sevres, Paris. The unit of mass is the kilogram, kept by the Bureau of Weights and Measures. Base Units in the Metric System The types of measurement you might need to know are given below. The first three are the most used in everyday life and the ones you need to learn. Length metre m Mass kilogram kg Time second s Temperature kelvin K Electricity ampere A Chemical quantities mole Mol Light candela cd Note that the common measurement of liquid volume, the litre, is not an international standard (SI) unit. However, because of its general application, it is still acceptable for use. 1 L = 0.001 m 3 Hence, 1 cm 3 = 1 ml In medical literature you may meet the abbreviation 'cc' for cubic centimetre. Study Guide Topic 8 Measurement

139 Prefixes To have some flexibility with measurement, we have some ways to describe parts of a metre or litre or kilogram. To make them easy to work with they increase or decrease in tens (although the official international standard prefixes go up in thousands). Here are some common prefixes: Prefix Definition Abbreviation Power of Ten Value micro a millionth μ (also mc) 10 6 1 1 000 000 milli a thousandth m 10 3 1 1000 Used for measuring small things centi a hundredth c 10 2 1 100 kilo thousands of k 10 3 1000 mega millions of M 10 6 1 000 000 giga billions of G 10 9 1 000 000 000 Used for measuring large things Want to know more prefixes? Go to: www.mathsisfun.com/measure/metric-system.html TEP023 Foundation Mathematics

140 Converting Units Length The standard length is the metre. Smaller and larger length units are related to the metre using prefixes. Below is a list of commonly used conversion factors. 1 m = 1000 mm 1 m = 100 cm 1 km = 1000 m This table includes commonly used lengths. Measurements like micrometres or megametres are correct but not commonly used. This diagram may help you to convert from one unit to another. 1000 1000 mm 10 10 cm 100 100 m 1000 1000 km 1000 1000 When you convert a large unit into a smaller unit you multiply. When you convert a small unit into a larger unit you divide. Study Guide Topic 8 Measurement

141 Example 1: Convert 367 cm to m Get the facts: converting cm to m (100 cm = 1 m) 100 ` cm m Are you multiplying or dividing? You are converting from smaller to larger means you divide. Step 3: 367 100 367 cm = 3.67 m Mass and Volume A Mg (megagram) equals 1000 kg, which is commonly called a tonne (abbreviation, t). This table includes commonly used weights and volumes. mcg mcl 1000 1000 mg ml 1000 1000 g L 1000 1000 kg kl 1000 1000 Mg ML Time Conversions You will also need to know the following time conversions Multiply Divide 60 60 24 7 52 TIME seconds minutes hours days weeks years There are 365 days in most years; leap years have 366 days. There are also close enough to 52 weeks in every year. TEP023 Foundation Mathematics

142 Example 2: Convert 6.1 kg to g Get the facts: converting kg to g (1 kg = 1000 g) 1000 ` g kg Are you multiplying or dividing? You are converting from larger to smaller means you multiply. Step 3: 6.1 x 1000 6.1 kg = 6100 g Example 3: Convert 2 338 100 ml to kl Get the facts: converting ml to kl (1000 ml = 1 L and 1000 L = 1 kl) 1000 1000 ml L kl Are you multiplying or dividing? You are converting from smaller to larger means you divide. Step 3: 2338100 1000 2 338 100 ml = 2338.1 L (half way there) 2338.1 1000 2338.1 L= 2.3381 kl Can you see how to do the two parts in step 3 in one go? Study Guide Topic 8 Measurement

143 Rates To this point we have compared things that are the same (ratios). Rates compare quantities that have different units. Some examples are: speed of a car 40 km/h (kilometres per hour) price of apples $3.50/kg (dollars per kilogram) speed of a drip 5 ml/min (millilitres per minute) Simplifying rates is the same as ratios, multiply or divide both sides by the same number Units must be shown in the final answer Rates are usually expressed by finding out how many of the first quantity correspond to one (per) of another Rates can be shown in different ways: eg. kilometres per hour = km/h = kmh -1 = kmph Calculating Rates The word per together with the rate unit gives the instruction for how to calculate a rate. For example, consider speed, km/h km / h distance time This means that if you know the distance (in kilometres) and the time taken (in hours) then the speed is calculated by dividing the distance by the time. TEP023 Foundation Mathematics

144 Example 1: I drove from Katherine to Darwin in 2.5 hours and covered 290 km. What was the speed (in kilometres per hour) over the whole trip? This means finding out the number of kilometres travelled in one hour. 290 km per 2.5 hours? km per 1 hour To work out the number of kilometres travelled in 1 hour we need to divide by 2.5 290 : 2.5 2.5 2. 5? : 1 Step 3: Step 4: Divide the kilometres by 2.5 also. Speed (in kilometres per hour) = 290 2.5 = 116 Answer the question: The speed was 116 km/h. Example 2: I bought 15 kilograms of meat and it cost $118.50. What did each kilogram cost? This means finding out the cost of the meat for 1 kilogram. If we put the words into mathematical language it would look like this: $118.50 : 15 kg $? : 1 kg To work out the cost of 1 kg of meat, we need to reduce 15 down to 1. 15 118.50 : 15? : 1 15 Step 3: Step 4: Divide the money by 15 also. $118.50 15 = $7.90 Answer the question: The meat cost $7.90/kg. Study Guide Topic 8 Measurement

145 Example 3: Convert the speed of a leak in the water tank from 2 L per minute to ml per second. Step 3: This means converting 1) Litres to millilitres (1 Litre = 1000 ml) 2 L = 2 1000 = 2000 ml 2) Minutes to seconds (1 minute = 60 seconds) 2 L : 1 min 2000 ml : 60 s To work out the number of ml per second we need to reduce 60 s down to 1 s. Divide the millilitres by 60 also. 60 2 000 ml : 60 s 60 33.3.. ml : 1 s 2000 60 = 33.333333... Step 4: Answer the question to a reasonable number of decimal places 33.33 ml/s TEP023 Foundation Mathematics

146 Scientific Notation Scientists and mathematicians often work with very small and very large numbers and so have a method of abbreviation called scientific notation. Scientific notation makes the number more compact by using powers of ten. It simply writes numbers in a shorthand form using exponents (powers) of ten. Very Large Numbers Example 1: 1 000 000 = 10 10 10 10 10 10 = 10 6 5 000 000 = 5 1 000 000 = 5 10 6 This is related to the number of places the decimal point has to move left or right original number = new number 10 to some power This new number is always a number between 1 and 9. Example 2: Write 1 256 000 000 in scientific notation Need to find how many places the decimal place at the end of this number (we don t write it but it is always there), would have to make to get up to just after the 1? 1256 000 000. There are 9 places to move the decimal point from the end of the number to the 1. Now we use powers to describe this. Step 3: 1 256 000 000 = 1.256 000 000 10 9 = 1.256 10 9 Decimal point is moved 9 places. Study Guide Topic 8 Measurement

147 Example 3: Write 8200 in scientific notation How many places would the decimal place at the end of this number have to make to get up to the 8? 8 200 There are 3 places to move the decimal point from the end of the number to the 8. Now we use powers to describe this: 8200 = 8.2 10 3 Decimal point is moved 3 places. Very Small Numbers If we have an awkward number like 0.000 000 942 we move the decimal point seven places to the right and the number becomes: NOTICE that the power is 9.42 10-7 negative here. Small numbers will have negative powers. Example 4: Write 0.000 002 3 in scientific notation How many places would the decimal place have to move to sit behind the first non zero number, 2 in this case? 0.000 002 3 There are 6 places to move the decimal point to after the 2. Now we use powers to describe this 0.000 002 3 = 2.3 10-6 Decimal point is moved 6 places left. TEP023 Foundation Mathematics

148 Converting from Scientific Notation Example 5: Write 6.51 x 10 5 in decimal notation The index or power is a positive number so the final answer will be a large number, the decimal point will have to move to the right. 6.51 6.51 10 5 = 651 000 Fill the empty spaces with zeros. Example 6: Write 4.951 x 10-6 in decimal notation The index or power is a negative number so the final answer will be a small number, the decimal point will have to move to the left. 4.951 4.951 10-6 = 0.000 004 951 Study Guide Topic 8 Measurement

149 Perimeter The distance around the outside of a figure is called the perimeter. It is measured in the units of length. In most cases, finding the perimeter of a simple shape is just a matter of adding the length of all the sides. Example 1: Find the perimeter. 7 m 8.5 m 9 m 11 m P = 8.5 + 7 + 9 + 11 = 35.5m Circumference of Circles The perimeter of a circle has its own name, the circumference. Let s have a look at the terminology associated with circles: The radius is the distance from the centre to the circumference. The diameter crosses from edge to edge going through the centre. Can you see that the diameter equals twice the radius? TEP023 Foundation Mathematics

150 The circumference of a circle is given by the formula: C = d or C = 2 r where d = diameter of the circle r = radius is an infinite decimal number that begins 3.14159... If you have a key on your calculator, use that value for area calculations. If not, approximate to 3.14. Example 4: Calculate the perimeter of this semicircle: 60 mm Find the circumference of a complete circle with a radius of 30 mm C 2 r = 2 30 188.5 Perimeter = 21 circumference of circle + diameter 21 188.5 + 60 154.3mm Study Guide Topic 8 Measurement

151 Example 5: A rope is wrapped around a circular pipe of diameter 25 cm, 6 times. Approximately how much rope is used? Give your answer to the nearest metre. Step 3: Find the amount of rope used for one coil in metres C = d = 25 78.54 cm 0.79 m Amount of rope needed for 6 coils Length = 6 0.79 = 4.71 m Answer the question: To the nearest meter, 5 meters of rope is needed. Perimeter of Triangles Perimeter = a + b + c If the triangle is right angled you can use Pythagoras Theorem to find any unknown sides. c 2 2 c a b a Not sure? Go to the end of Topic 7 for a reminder of Pythagoras Theorem. b TEP023 Foundation Mathematics

152 Example 3: Calculate the perimeter of this polygon 6.5 m 4 m A polygon is a shape with many sides (poly meaning many). 3.5 m 8 m Need to find the missing side using Pythagoras. 3 m d Finding the vertical 6.5 3.5 = 3 Finding the horizontal 8 4 = 4 4 m d 2 2 2 3 4 d d 5 2 4 3 4 Now you can add up all the sides. P = 6.5 + 5 + 4 + 3.5 + 8 = 27 m Study Guide Topic 8 Measurement

153 Area The area of a shape is a measure of the surface enclosed by the shape. You want to work out how many unit squares cover that surface. As you will see below, formulas for finding the area of shapes are a useful shortcut. This is a square centimetre box, which equals 1 cm 2. 1 cm 1 cm Example 1: Find the area of the rectangle given below with sides of length 5 m and 3 m. 5m 3m You could count the number of squares (of 1m 1m) that make up the rectangle. You should count 15 squares so your answer would be written as 15 m 2. (m 2 is short hand for square metres).or you could look for a short cut. Area of a rectangle can be found using the formula: Area = length width. A lw Step 3: A 5 3 =15 m 2 TEP023 Foundation Mathematics

154 Areas of Some Common Shapes Square Area = L L = L 2 L L Rectangle Area = LW W L 1 Triangle Area = b h 2 h b Circle Area = r 2 r Study Guide Topic 8 Measurement

155 Example 2: Find the area of a circle with radius 10 cm. A = r 2 = 10 2 100 = 314.2 cm 2 Example 3: Find the area of a circle with diameter 30 cm Since diameter = 2 radius then r = 30 2 So A = = 15 2 r 2 = 15 706.9 cm 2 TEP023 Foundation Mathematics

156 Composite Areas Sometimes it is necessary to put in a construction line to divide a shape up into shapes you know the areas of. Example 4: Find the area of the shape below. 4m 3 m This shape is made up of a square and a triangle. You need to find the area of the triangle and the square. Area of a triangle = ½ base height 1 Atriangle b h 2 = 1 3 1 2 = 1.5 m 2 Step 3: Area of square = L 2 A = 3 3 = 9 m 2 1 3 Step 4: Add the two areas Area = 1.5 + 9 = 10.5 m 2 Study Guide Topic 8 Measurement

157 Example 5: Calculate the area of glass in this window. 2 m Divide the shape into its components This is a semi-circle of radius 1 m. This is a square with side lengths of 2 m. 2 m Find the area of each component: Area square = L 2 = 2 x 2 = 4 m 2 Area semicircle = = 1 2 r 2 1 2 1 2 1.57 m 2 Step 3: Add the areas of the component parts Area total = 4 + 1.57 = 5.57 m 2 2 m TEP023 Foundation Mathematics

158 Example 6: Find the area of a 1 m path around a rectangular garden 6 m by 3 m. Draw the shape 6m 3 m 1 m Step 3: Step 4: Area of total shape (path and garden) = 8 5 = 40 m 2 Area of garden = 6 3 = 18 m 2 Area of path = Area of path and garden Area of garden = 40 18 = 22 m 2 Study Guide Topic 8 Measurement

159 Example 7: Find the area of the paddock below. Give your answer in the following units: (i) km 2 (ii) m 2 (iii) ha (hectares) 1ha = 10 000m 2 1.7 km 0.9 km Choose your formula Area = length width = 0.9 1.7 = 1.53km 2 The question asks for the answer in metres, so change the kilometres to metres before using the formula to calculate the area. 1.7 km = 1700 m (since 1000 m = 1 km) 0.9 km = 900 m Area = length width. = 900 1700 = 1 530 000 m 2 Step 3: Hectares (ha) : 1ha = 10 000m 2 Area = 1 530 000 10 000 = 153 ha TEP023 Foundation Mathematics

160 Converting Units of Areas A little caution must be used when converting to multiples or submultiples of the base unit. Consider the following square, of 1cm by 1cm. Each side is divided into ten sections; each section is 1mm long. 1 cm 1 cm The area of each small square is therefore 1 mm 2 ; the area of the large square is 1 cm 2. How many square millimetres are there in 1 cm 2? By counting, it can be seen that there are one hundred: 1 cm 2 = 100 mm 2 or 10 2 mm 2 When converting units of area we must square the linear conversion factor as well as the units. Example 8: Convert 1 m 2 to square centimetres. 1 m = 100 cm 1 m 2 = 100 2 cm 2 = (100 100) cm 2 = 10 000 cm 2 Convert 1 m 2 to square millimetres 1 m = 1000 mm = 1000 2 mm 2 = (1000 1000) mm 2 = 1 000 000 mm 2 Study Guide Topic 8 Measurement

161 Volume The volume of a shape is a measure of the amount of space inside it. 1 m 1 m 1 m You want to work out how many unit cubes are needed to fill the figure. As you will see below, formulas for finding the volume of shapes are a useful shortcut. Calculating the Volume of a Box Imagine a layer of blocks arranged in two rows of five. The area covered by this layer = 2 x 5 = 10 Now stack up three layers of these blocks. TEP023 Foundation Mathematics

162 You would end up with a stack that looks like this. Your stack is now a box with 3 layers of 2 by 5 blocks, so its volume is: V = 2 5 3 = 30 cubic units Volume of Prisms A prism is a solid object that has the same cross section along its length. The end faces are the same shape and size and are parallel. The shape of the ends give the prism a name. This is triangular prism. This is rectangular prism. Volume of rectangular prism = length width height V lwh Volume of a prism = area of the cross section height of the prism V A h Study Guide Topic 8 Measurement

163 Example 1: Find the volume of this rectangular prism. 80 cm 100 cm 90 cm You could count the number of cubes (of 1cm 1cm 1cm) that make up the cube or you could look for a short cut. Volume of rectangular prism is found using the formula: Volume = length width height V l w h Step 3: V 100 90 80 = 720 000 cm 3 Example 2: Find the volume of the triangular prism. Find the area of the cross section, ie the triangular end. A = 1 2 bh = 1 14 16 2 =112 cm 2 Find the volume V = A h = 112 30 = 3360 cm 3 TEP023 Foundation Mathematics

164 Example 3: Find the volume of the cylinder with radius 10 m and height 550 cm. h r Step 3: Make sure units are all the same!! Change all units to metres 550 cm = 5.5 m So, r = 10 and h = 5.5 Find the area of the base. In this question, the base is the circular end A = r 2 = 10 2 314 m 2 Find the volume V = area of base h = 314 5.5 = 1727 m 3 Study Guide Topic 8 Measurement

165 Volume of Pyramids A pyramid is a solid object where: The base is a polygon (a straight-sided shape) The sides are triangles which meet at the top (the apex). You might think that pyramids are more complicated than prisms, but finding their volume is easy. A pyramid is placed inside a prism as shown. How much volume do you think the pyramid will hold compared to the prism? If you guessed that the pyramid would hold one third of the prism, then you were correct. The volume of a pyramid is one third the volume of its corresponding prism. V pyramid = = 1 area of base height 3 1 3 A base h TEP023 Foundation Mathematics

166 Example 4: Find the volume of the pyramid below. 60 m 15 m 55 m Find the area of the base A = lw = 60 15 = 900 m 2 Find the volume V = 1 3 Ah = 1 3 900 55 =16 500 m 3 Study Guide Topic 8 Measurement

167 Units of Volume The international standard base unit of volume is the cubic metre (m 3 ). To convert between multiples and submultiples of the base unit, consider the following cube which is 1cm 1cm 1cm. If each side of the cube is divided into ten intervals (each 1mm), we find that 1000 cubic millimetre units are contained in the cube. 1 cm = 10 mm 1 cm 3 = 10 3 mm 3 = 1000 mm 3 Similarly, 1 m 3 = 100 3 cm 3 = 1 000 000 cm 3 1 m 3 = 1000 3 mm 3 = 1 000 000 000 mm 3 When converting units of volume we must square the linear conversion factor as well as the units. Example 1: Convert 3 m 3 to cubic centimetres 3 m 3 1 000 000 = 3 000 000 cm 3 Example 2: Convert 15 609 cm 3 to cubic metres 15 609 cm 3 1 000 000 = 0.015 609 m 3 TEP023 Foundation Mathematics

168 Study Guide Topic 8 Measurement