3. NUMBERS IN MEASUREMENTS

• At the end of this outcome, you will:• be able to read measurements

• be able to estimate measurements

• use formula to calculate measurements

• be able to convert measurements between different units

Units of measurementMeasuring unit Medium

Distance Metre (m)

Area Square metre (m2)

Volume Cubic metre (m3)

Volume Litre

Activity 17

1. Convert the quantities in the first column to the units in the second column:

450m km 24,051kg g

32.8kg m 31km m

8,257l ml 60kl L

749,5g kg 141g mg

3824ml l 92m mm

2575mm m 79,5l ml

0,45

32 800

8257

0,7495

3,824

2,575

24051

31000

60000

141000

92000

79500

Activity 17

2. Complete the following sentences:a. A millionth of a metre is called ab. A thousand metres is called ac. A hundredth of a metre is called ad. Write the value of one teragram, in other words,

one teragram = ge. In computer terminology one megabyte implies

bytesf. One picometre equals metreg. What is 10-9 called?h. One ns is equal to of a secondi. One MW is equal to Watts

millimetre

kilometre

centimetre

1 000 000 000 000

1 048 576

0.000 000 000 001

1 nano litre

0.000 000 001

1 000 000

Activity 17

3. Select the correct measuring unit and instrument to complete the table: Kitchen scale, tape measure, measuring jug, pipette, burette, trundling wheel, vernier calliper, water displacement method, municipal water meter, bathroom scale

Dimension to be measured Unit of measurement

Measuring instrument

Distance run on a marathon route

Fluid measured to exact millimetre amount by sucking out of a container

Internal diameter of a pipe

Weight of blue marlin caught from a deep-sea fishing boat

Mass of a person before and after going on a diet

Mass of ingredients used to bake a cake

Mass of a ship’s container

Size of a virus through a microscope

Amount of water used by a municipality during 24 hours

Fluid measure in a laboratory by opening a small tap in a glass tube to add drop by drop to a mixture

Time slept during a lifetime of 60 years

Time necessary to boil water for sterilisation purposes

Trundling wheelkilometres

millilitre Pipette

millimetre Vernier caliper

kilogram

kilogram Bathroom scale

grams Kitchen scale

tons

Nano metres Microscope

Kilolitres Municipal water meter

Millimetre Burette

Years Calendar

Minutes Clock

3.2 Measuring instruments

• Length– Tape measure

– Ruler

– Vernier caliper

– Screw micrometer

3.2 Measuring instruments

• Volume– Household water usage is measured in kilolitres

– Petrol poured into a tank is measured in litres

– Injection needles are measured in millimetres

3.2 Measuring instruments• Weight

– Weight is the common name for the gravitational force of an object

– It is measured in Newton like all other forces. An astronaut with a mass of 70 kg has a weight of 70 X 9,8 ≈ 690 N on earth.

– That is the strength of the earth’s gravitational pull on the astronaut’s 70 kg of mass. But on the moon his weight will only equal 110 N because the moon has less force to attract the astronaut with. On a massive planet like Jupiter, the astronaut would weigh more than five

– times as much as on earth, i.e. 3600 N. Wherever the astronaut stands, he will still have the same amount of matter, which is measured as 70 kg.

– A kilogram bag of sugar has a mass of 1 kilogram, and a weight on earth of 9,8 N. Most scales give a reading in mass units such as kilograms.

3.2 Measuring instruments

• Mass– This is a measure of the amount of matter in a

substance. Mass is measured in kilograms (kg).

– The mass of a truck or a ship is given in metric ton.

– Your mass is measured in kilograms on a bathroom scale, that of a frog in grams on the kitchen scale and that of a spider in milligrams on a chemical balance.

3.2 Measuring instruments

• Temperature– Temperature is measured with a thermometer in

degrees Celsius or centigrade (ºC) or in degrees Fahrenheit (Imperial system) or in Kelvin.

– Freezing point of water is 0ºC or 32ºF or 273 K.

– Below zero measurement of temperature is normally given in negative units of degrees Celsius.

– Boiling point of water at sea level is 100ºC or 212ºF.

3.2 Measuring instruments

• Gravitation– The force of attraction between masses is known as

gravitational force. – At a given location on earth and in the absence of

air resistance, all dropped objects will fall with the same constant acceleration.

– We call this the acceleration due to gravity of the earth at that particular place on earth, and we give it the symbol “g”. Its magnitude is approximately:

– g = 9,80 m / s². It is not exactly the same everywhere. It is called the gravitational constant.

3.2 Measuring instruments

• Force– This is an action capable of accelerating an object.

It is the amount of push or pull to speed up the object. It is measured in Newtons. One Newton is the force required to impart an acceleration of 1 m / s² to a mass of 1 kg. Newton is mass times acceleration.

– Thus 1 N = 1 kg . m / s².– When we speak of the weight of something it is a

force towards earth. – Weight = mass times the gravitational constant.

Weight = mass x 9,80 N

3.2 Measuring instruments

• Speed– Speed is a measure of the time taken to travel a

certain distance. It is also called a rate of change. If you run 40 km in 4 hours, then it means that in every hour you run 10 km, or that you run 10 km/h. To get to the ten km/h you had to divide the 40 km by the 4 hours.

– Therefore, speed is calculated by dividing the distance travelled, by the time taken to travel the distance:

– Speed = distance time

3.2 Measuring instruments

• Velocity– The velocity of a moving object is given by

stating both its speed and the direction in which the object is moving. If the velocity is stated without reference to any direction, it must be assumed that the object is travelling in a straight line and that the overall direction of that line does not matter.

Time

Dis

tanc

e

Zero velocityObject at rest

Constant

Velocit

y

3.2 Measuring instruments• Acceleration

– The acceleration of a moving object is a measure of how its velocity/speed changes in relation to time.

– Acceleration may be positive (for speeding up), or negative (for slowing down).

– The SI unit of acceleration is metres per second per second, abbreviated to m/s2.

– Acceleration equals difference in speed divided by difference in time. It is the gradient of the speed function or in other words it is the second derivative of the function: distance against time.

– A velocity-time graph is a graph that shows the relationship between the velocity of an object in relation to time. The gradient of a line drawn on the graph is a measure of acceleration. If the line of the relationship is curved, the particular acceleration at some moment in time can only be found by drawing a tangent to the curve at that point and measuring its gradient. The area under the curve between two ordinates drawn from the time scale, measures the distance travelled in that time interval.

Velocity versus time

Conversion of speed units

m.h-1 km.h-1 m.s-1 ft.s-1

m.h-1 1,6093 0,44704 1,4667

km.h-1 0,62137 0,27778 0,91134

m.s-1 2.2369 3,6 3,2808

ft.s-1 0,68182 1,0973 0,3048

Activity 18

2. Read the following scales and give the answers with the correct abbreviations of measurement

a. Volume

a.0.5mlb.5mlc.1,05mld.1,5mle.1,9 ml

b. Mass

c. length

a.0,30 kgb.0,90kgc.1,50kgd.1,95kge.0,05kgf.0,35kgg.0,55kgh.0,90kg

i.0,2 kgj.0,7kgk.1,4 kgl.1,9 kgm.0,25kgn.0,63kgo.1,12 kgp.1,75kg

Activity 183. There are many different rates

a. If you travelled 2 500 km in 16 hours, at what speed did you travel?

b. A cyclist completed 42,5 km in 2 hours and 15 minutes. At what speed did he travel? Remember that you should work only in hours to give an answer in km/h.

c. A flight of 6 000 km takes 22 hours. The aircraft stops for refuelling twice for a total of 45 minutes. At what speed did the aircraft fly?

d. A typist types 600 words in 27 minutes. What speed is she typing at?

e. An employee takes two and a half hours at KWV to bottle 7,5 kilolitres of wine at his station in the bottling chain. What is the bottling speed in litres per hour?

156,25 km /h

18.88 km/h

282,35km/h

22.2 words per minute

3000 litres per hour

Activity 18

3. There are many different ratesf. A seamstress completes 6 dresses in 8 hours. At

what speed is she working (dresses per hour)

g. Interpret the following graphs of distance against time and speed against time

0,75 dresses per hour

Constant speed to point A. Stationary from A to B. Constant speed to point C

Activity 18

3. There are many different ratesh. Water leaks from a tank at the rate of 5ml per second. If

the tank has 1950ml of water, how long will it take before it is empty?

i. A train travels 144km in two hours. Find the speed of the train in

a. km /hb. m /s

j. Which is the better rate for a tourista. R2490 for 3 days, or R2790 for 4 days at a Safari Lodge in

Botswanab. R2 850 for 4 days, or R3 590 for 5 days at a Victoria Falls Hotel

390 seconds = 6 ½ minutes

72 km/h144 x 1000 ÷ 3600 = 40m/s

R830 per day vs R697,50. 4 days is better value per day

R712.50 vs 718; 4 days is better value for money per day

Activity 19• At the end of December your mother or friend’s mother receives a water and

electricity bill of over R800-00 where she normally pays more or less R350-00. She asks you to investigate her water usage and to advise her on ways to save water. Locate the water meter and take the meter reading. There are three different scales of units on this meter.

1. Find out from your local municipality which measurements of volume the different scales represent.

2. See whether the reading remains constant when no taps are turned on. If the meter ticks on steadily even though no taps are open, find out where the leak in the system is. If the reading remains constant, give a reason for the elevated water usage.

3. Investigate water usage in the sprinkler irrigation lines in the garden. Turn them on separately for 20 minutes and for each twenty minute period, record the meter reading. See if there are any leaks in the system, e.g. has a garden fork pierced a hole in the hose? Calculate how much water will be saved by watering only every second day.

Activity 19

4. When next the washing machine is used on a full load, measure the water usage with no other taps running. In a similar fashion establish how much water is used in the toilet, the shower and the bath. Draw a table of the results.

5. Discuss possible methods of saving water/ reducing water wastage. Take into account the frequency of watering the garden, the washing of dishes or clothes, water usage during showers and baths. Plan the household’s activities to keep the water bill down to R450-00. Price of water varies on a sliding scale according to the volume used.

Activity 19

• Use the table below to:

a. Do the calculations with both the 1005 as well as the 2008 sliding indices

b. Calculate the % difference between 2005 and 2008 in the price of the total amount of water used

Kilolitres used

Cost per kilolitre/2005

0-6 No cost

7-10 R1,41

11-30 R2,32

31-56 R3,95

57-80 R4,46

Above 80 R6,00

Kilolitres used

Cost per kilolitre/2008

0-6 No cost

7-10 R1,95

11-30 R3,88

31-56 R6,23

57-80 R8,04

Above 80 R10,83

3.3 Using a formula to calculate a measurement

Shape Formula for perimeter

Triangle, trapezium and any other odd shape

Add the length of the sides. If some lengths are not given, you will be able to calculate the distances from indirect measurements

Square 4 times side length = 4l = 4s

Rectangle 2 lengths + 2 breadths = 2l + 2b

Circle 2 r

3.3 Using a formula to calculate a measurement

2-D shape2 Formula for area

Rectangles l x b

Parallelogram B x H

Trapezium h/2 (P1 + P2)

Triangle ½bh

Circle r2

3.3 Using a formula to calculate a measurement

3-D object Formula for volume

Rectangular solid length x breadth x height

Cylinder r2h

Activity 20

1. Calculate the perimeter of the rectangles with the following dimensions:

a. length = 12cm; breadth = 8cm

b. length = 45mm; breadth = 2,6 cm

c. length = 16,25m; breadth = 1238cm

40cm

142 mm

5726cm

Activity 20

2. Now calculate the area of the rectangles with the dimensions in the previous question (Use the formula: area of a rectangle = length times breadth)

1.96cm2

2.1179cm2

3.201.175m2

Activity 20

3. In a fruit juice processing factory, the different juice types have to be kept at different temperatures for different time periods to ensure complete sterilisation. The constant temperature pipe through which the juice flows, is: 5 metres long for pear juice (juice remains in the pipe for 45 seconds), and 6 metres long for tomato juice (juice remains in the pipe for 72 seconds).

• Calculate the speed through the pipe system of the two different juices.

Pear juice: 0,11 m /sTomato juice: 0,083 m/s

Activity 20

4. a. Measure the diameter of a R5,00 coin to the nearest

centimetreb. Then, measure the diameter of the coin to the nearest

millimetrec. Which measurement is more accurate?d. When the coin is measured to the nearest centimetre what is

the greatest possible error? e. When the coin is measured to the nearest millimetre, what is

the greatest possible error? (The precision of the measurement is always the same as the unit used to measure, and the greatest possible error is always half of this unit)

3cm

26mm

Measuring to millimetres

0,50 cm

0,5 mm

Activity 20

5. a. If something is correctly measured as 5 m to

the nearest metre, what is the smallest it could be?

b. How much shorter than 5 m could it be? Explain why?

4,5m

0,5m

Activity 20

6. You have to calculate the area of stainless steel sheeting necessary to make a cylinder which will be used as a fermentation tank in the making of wine. The cylinder has to have a capacity of 230 000 litre. The radius of the tank is 1,5 m. Use the different values for given in the example above and compare the answers for total external surface area. Which value would you use to order the stainless steel sheeting? – Formula for total external surface area of a cylinder = 2r2 +

2rh– Formula for the volume of a cylinder = r2h

Radius of 1,5 metres

Activity 20

7. You have to order the correct volume of paint for the outside surface of a cylinder. The cylinder has a radius of 1,5 m and a height of 3,25 m.

• Use a value of 3,1519 for and calculate how much paint to order.

• Formula for total external surface area of a cylinder = r2 + 2rh.

2(1,5)(1,5 + 3,25) = 44.77m2

Activity 21• A manufacturer wants to sell fruit juice in cylindrical

one-litre cans, using the least possible amount of metal. For the litre can, he has to find the radius and height which will give the smallest surface area of metal.

• Work to the nearest millimetre and approximate to the third decimal place for height of the cylinder and to the second decimal place for the total external surface area of the cylinder.

• Use the formulae:– Volume of a cylinder = r²h– Total external surface area of cylinder = 2r² + 2rh

Activity 21

a. Complete the table by calculating the total external surface area of the cans

b. Decide on dimensions for the cylinder which will use the least metal.

c. If you had worked to the nearest centimetre and had approximated all calculations to one decimal place, calculate how the answer you chose would have differed

Radius (cm) 1 2 3 4 5 6 7 8 9 10

Height of can (cm)

Tot ext surf area

318,310 79,577 35,368 19,894 12,732 8,842 6,496 4,974 3,930 3,183

2006,28 1025,13 723,21 600,52 577,08 559,53 593,59 652,12 731,12 828,32

6cm