Measurement and energy - web2.hunterspt-h.schools.nsw.edu.auweb2.hunterspt-h.schools.nsw.edu.au/studentshared/MATHEMATICS… · Chapter 4 . Measurement and energy. 4A. Units of measurement.

4

Measurement and energy

Syllabus topic — M1.1 Practicalities of measurement

M1.3 Units of energy and mass

This topic provides students with the opportunity to appreciate inherent errors in

measurements and to become competent in solving practical problems involving energy.

Outcomes

• Review the use of different units of measurement.

• Calculate the absolute and percentage error in a measurement.

• Use standard form and standard prefixes in the context of measurement.

• Express numbers to a certain number of significant figures.

• Use units of energy and mass related to food and nutrition.

• Use units of energy to describe consumption of electricity.

• Investigate common appliances in terms of energy consumption.

Digital Resources for this chapter

In the Interactive Textbook:

• Videos • Literacy worksheet • Quick Quiz

• Solutions (enabled by teacher) • Widgets • Study guide

In the Online Teaching Suite:

• Teaching Program • Tests • Review Quiz • Teaching Notes

Knowledge check

In the Interactive Textbook you can take a test of prior knowledge required for

this chapter, and depending on your score you may be directed to revision from

the previous years’ work.

Mathematics Standard Year 11 ISBN 978-1-108-43463-8 © Powers 2018 Cambridge University PressPhotocopying is restricted under law and this material must not be transferred to another party.

Chapter 4 Measurement and energy128 4A

Units of measurement

Measurement is used to determine the size of a quantity. It usually involves using a measuring instrument. For example, to measure length, instruments that can be used include the rule, tape measure, caliper, micrometer, odometer, laser range finder, and GPS. There are a number of systems of measurement that define their units of measurement. We use the SI metric system.

SI unitsThe SI (‘Systeme International’, or International System of Units of Measurement), is based on multiples of 10. It is a version of the metric system that allows easy multiplication when converting between related units. Units shown in black are SI and those in red (below) are non-SI units approved for everyday or specialised use alongside SI units.

Quantity Name of unit Symbol Value

Length MetreMillimetreCentimetreKilometreNautical mile

mmmcmkmnm

Base unit1000 mm 1 m=100 cm 1 m=1 km 1000 m=1 nm 1852 m=

Area Square metreSquare centimetreHectare

m2

cm2

ha

Base unit=10 000 cm 1 m2 2

=1 ha 10 000 m2

Volume Cubic metreCubic centimetreLitreMillilitreKilolitre

m3

cm3

LmLkL

Base unit=1000 000 cm 1 m3 3

=1 L 1000 cm3

1000 mL 1 L=1 kL 1000 L=

Mass KilogramGramMilligramTonne

kggmgt

Base unit1000 g 1 kg=1000 mg 1 g=1 t 1000 kg=

4A

MassMass is a measure of the amount of matter in a body. There is a difference between mass and weight. Weight is the measure of the amount of force acting on the mass due to gravity. However if you’re on the surface of the Earth and not moving, mass and weight can be considered to be the same in everyday contexts. If you change your location with respect to gravity, mass will remain unchanged, but weight will not.

Converting between SI units of the same type

The SI prefix to the unit indicates the conversion between units of the same type. It indicates a

multiple of 10. Common prefixes are mega (1000 000), kilo (1000), centi 1100 )( and milli 1

1000 )( .


129 4A Units of measurement

Converting area and volume unitsTo convert area units, change the side length units and compare the values for area.

1 m

1 m 100 cm

100 cm== × =1m 100 100 10 000 cm2 2

=1 m 10 000 cm2 2

or =1 cm 110 000

m2 2

To convert volume units, change the side length units and compare the values for volume.

1 m

1 m 100 cm100 cm

100 cm=1 m

= × × =1 m 100 100 100 1000 000 cm3 3

=1 m 1000 000 cm3 3

or =1 cm 11000 000

m3 3

Example 2: Converting units of mass 4A

Complete the following.a 5.2 kg g= b 3850 kg t=

SOLUTION:

1 To change kilograms to grams multiply by 1000.

2 To change kilograms to tonnes divide by 1000.

a 5.2 kg 5.2 1000 5200 g= × =

b 3850 kg 3850 1000 3.85 t= ÷ =

Example 1: Converting units of length 4A

Complete the following.a 35cm mm= b 4500 m km=

SOLUTION:

1 To change cm to mm multiply by 10.

2 To change m to km divide by 1000.

a 35 cm 35 10 mm

350 mm

= ×=

b = ÷=

4500 m 4500 1000 km

4.5 km

LENGTH, MASS AND VOLUME

To change from one unit to another, multiply or divide by a power of 10.

unit

kilo× 1000

× 1000

× 100

× 10

÷ 1000

÷ 1000

÷ 100

÷ 10centi

milli

mega



Exercise 4A LEVEL 1

1 Complete the following.a 5 cm mm= b 78 m cm= c 2 km m=

d 890 m cm= e 57 cm mm= f 6 km cm=

g 9400 m km= h 600 mm cm= i 8100 cm m=

j 49 000 cm km= k 22 000 m km= l 51mm cm=

2 Complete the following.a 3 t kg= b 45 kg g= c 76 t kg=

d 8100 kg g= e 4 t g= f 0.52 t kg=

g 6800 g kg= h 9 300 000 g t= i 45000 000 g t=

j 300 kg t= k 2300 g kg= l 60 000 g kg=

3 Complete the following.a 2 L mL= b 12 kL L= c 9 kL mL=

d 7800 kL L= e 50 L mL= f 300 kL mL=

g 6100 L kL= h 400 mL L= i 210 000 mL kL=

j 80 mL L= k 79000 mL kL= l 8000 000 mL kL=

4 What unit of length is most appropriate to measure each of the following?a Length of a penb Height of a building

c Thickness of a credit cardd Distance from Sydney to Newcastle

e Height of a personf Length of a football field

5 What unit of mass is most appropriate to measure each of the following?a Weight of an elephantb Mass of a mug

c Bag of onionsd Weight of a baby

e Mass of a truckf Mass of a teaspoon of sugar

Example 1

Example 2


131 4A Units of measurement

6 There are 20 litres of a chemical stored in a container.a What amount of chemical remains if

750 mL is removed from the container? Answer in litres.

b How many containers are required to hold a kilolitre of the chemical?

7 The length of the Murray River is 2575 km. The length of the Hawkesbury River is 80 000 m. What is the difference in their lengths? Answer in metres.

8 Christopher bought 3 kg of sultanas. What mass of sultanas remains if he ate 800 grams? Answer in kilograms.

9 A truck is loaded with 3000 bricks, each of mass 4 kg. Find the total mass (in t) of the bricks.

10 The mass of a box of cereal is 375 g. What is the total mass (in kg) of 15 boxes of cereal?

11 Find the total mass (in g) of five items of mass 250 mg,1100 mg, 0.7 g, 5.95 g, and 15.4 g.

12 An empty plastic box has mass 750 g. It is packed with 8 bottles of drink, each of mass 1.25 kg. Calculate the total mass (in g) of the packed box.

13 There are three tonnes of grain in a truck. What is the mass if another 68 kg of grain is added to the truck? Answer in kilograms.

14 Find the total mass (in kg) of the following contents of a shopping trolley:muesli, 600 g instant coffee, 250 g frozen peas, 1 kgrice, 3 kg self-raising flour, 2 kg strawberry jam, 400 g



LEVEL 2

15 The distance from John’s home to Bankstown is 15.9 kma John travels to Bankstown and back 5 times each week. What is the total distance travelled?

b John’s car can travel 8 km on 1 litre of petrol. How many litres of petrol will John use each week travelling to Bankstown and back?

16 A cyclist travels to and from work over a 1200-metre long bridge. Calculate the distance travelled in a week if the cyclist works for 5 days. Answer in kilometres.

17 Madison travels 32 km to work each day. Her car uses 1 litre of petrol to travel 8 km.a How many litres of petrol will she use to get to work?

b How many litres of petrol will she use for 5 days of work, including return travel?

18 Arrange 500 m, 0.005 km, 5000 cm and 5000 000 mm in:a ascending order (smallest to largest)b descending order (largest to smallest)

19 Complete the following.a 1 km m2 2= b 1 m mm2 2=

c 1 cm mm2 2= d 1000 cm m2 2=

e 2000 mm cm2 2= f 5000 m km2 2=

g 3.9 m cm2 2= h 310 km m2 2=

i 4.7 m mm2 2= j 74300m km2 2=

k 6500 mm cm2 2= l 4000 cm m2 2=

20 The area of a field is 80 000 square metres. Convert the area units to the following.a Square kilometres

b Hectares

LEVEL 3

21 Jackson swims 30 lengths of a 50-metre pool.a How many kilometres does he cover?

b If his goal is 4 kilometres, how many more lengths must he swim?


133 4B Measurement errors

Measurement errors

There are varying degrees of instrument error and measurement uncertainty when measuring. Every time a measurement is repeated, with a sensitive instrument, a slightly different result will be obtained. The possible sources of errors include mistakes in reading the scale, parallax error and calibration error. The accuracy of a measurement is improved by making multiple measurements of the same quantity with the same instrument.

Accuracy in measurementsThe smallest unit on the measuring instrument is called the precision or limit of reading. For example, a 30 cm ruler with a scale for millimetres has a precision of 1 mm. The accuracy of a measurement

is restricted to plus or minus half )(± 12

of the precision. For example,

if the measurement on the ruler is 10 mm then the range of errors is ±10 0.5 mm. Here the upper bound is +10 0.5 mm or 10.5 mm and

the lower bound is 10 – 0.5 mm or 9.5 mm.

Every measurement is an approximation and has an error. The absolute error is the difference between the actual value and the measured value indicated by the instrument. The maximum value

for an absolute error is 12

of the precision.

4B

1 cm 2 3 4 5

RELATIVE ERROR PERCENTAGE ERROR

±

Absolute errorMeasurement

±

×

Absolute errorMeasurement

100%

PRECISION ABSOLUTE ERROR UPPER BOUND LOWER BOUND

Smallest unit on measuring instrument or limit of reading

Measured value – Actual value

12

precision± ×

Measurement + Absolute error

Measurement - Absolute error

Relative error gives an indication of how good a measurement is relative to the size of the quantity being measured. The relative error of a measurement is calculated by dividing the precision by the

actual measurement. For example, the relative error for the above measurement is )( =0.510

0.05. The

relative error is often expressed as a percentage and called the ‘percentage error’. For example, the

percentage error for the above measurement is )( × =0.510

100 5%.


Chapter 4 Measurement and energy134 4B

Example 3: Finding the measurement errors 4B

a What is the length indicated by the arrow on this ruler?

b What is the precision or limit of reading?c What is the upper and lower bound for each measurement?d Find the relative error. Answer correct to three decimal places.e Find the percentage error. Answer correct to one decimal place.

SOLUTION:

1 The arrow is pointing to 38 mm.

2 Precision is the smallest unit on the ruler (millimetre).

3 Calculate half the precision.

4 Lower bound is the measured value minus 1

2 the precision.

5 Upper bound is the measured value plus 1

2 the precision.

6 Write the formula for relative error.

7 Substitute the values for absolute error and the measurement.

8 Evaluate correct to three decimal places.

9 Write the formula for percentage error.

10 Substitute the values for absolute error and the measurement.

11 Evaluate.

a Length is 38 mm.

b Precision is 1 mm.

c 12

precision 12

1

0.5 mm

× = ×=

Lower bound 38 0.5 37.5mm= - =

Upper bound 38 0.5 38.5mm= + =

d RelativeerrorAbsolute errorMeasurement

0.538

0.013

= ±

= ±

≈ ±

e PercentageerrorAbsolute errorMeasurement

100%

0.538

100%

1.3%

= ±

×

= ± ×

= ±

0 10 20 30 40 50 60 70 80

� ±1.3%


135 4B Measurement errors

Exercise 4B

1 Four measurements of length are shown on the ruler below.

0 10 20 30 40 50 60 70 80 90 100

A B C D

a What length is indicated by each letter? Answer to the nearest millimetre.

b What is the precision or limit of reading?

c What the largest possible absolute error?

d What is the upper and lower bound for each measurement?

e Calculate the relative error, correct to three decimal places, for each measurement.

f Calculate the percentage error, correct to one decimal place, for each measurement.

2 Two measurements of mass are shown on the scales below.

00.5

1kg

1.5

2kg2.5

3kg

3.5

4kg

4.5

A

00.5

1kg

1.5

2kg2.5

3kg

3.5

4kg

4.5

B

a What mass is indicated by A and B?

b What is the precision or limit of reading?

c What the largest possible absolute error?

d What is the upper and lower bound for each measurement?

e Calculate the relative error, correct to three decimal places, for each measurement.

f Calculate the percentage error, correct to one decimal place, for each measurement.

LEVEL 1

Example 3


Chapter 4 Measurement and energy136 4B

LEVEL 2

3 A dishwasher has a mass of exactly 49.6 kg. Abbey measured the mass of the dishwasher as 50 kg to the nearest kilogram.a Find the absolute error.

b Find the relative error. Answer correct to three decimal places.

c Find the percentage error. Answer correct to two decimal places.

4 An iPhone has a mass of exactly 251 g. Vivaan measured the mass of the iPhone as 235 g to the nearest gram.a Find the absolute error.


c Find the percentage error correct to two decimal places.

5 An LCD screen has a mass of exactly 2.71 kg. Saliha measured the mass of the screen as 3 kg to the nearest kilogram.a Find the absolute error.


c Find the percentage error correct to three decimal places.

6 A measurement was taken of a skid mark at the scene of a car accident. The actual length of the skid mark was 25.15 metres; however, it was measured as 25 metres.a What is the absolute error?


c Find the percentage error. Answer correct to one decimal place.

LEVEL 3

7 The length of a building at school is exactly 56 m. Cooper measured the length of the building to be 56.3 m and Filip measured the building at 55.8 m.a What is the absolute error for Cooper’s measurement?

b What is the absolute error for Filip’s measurement?

c Compare the relative error for both measurements. Answer correct to four decimal places.

d Compare the percentage error for both measurements. Answer correct to three decimal places.


137 4C Standard form and significant figures

Standard form and significant figures

Standard formStandard form or scientific notation is used to write very large or very small numbers more conveniently. It consists of a number between 1 and 10 multiplied by a power of 10. For example, the number 4 100 000 is expressed in scientific notation as ×4.1 106. The power of 10 indicates the number of tens multiplied together. For example:

× = × × × × × ×=

4.1 10 4.1 (10 10 10 10 10 10)

4 100 000

6

When writing numbers in scientific notation, it is useful to remember that large numbers have a positive power of 10 and small numbers have a negative power of 10.

4C

WRITING NUMBERS IN SCIENTIFIC NOTATION

1 Find the first two non-zero digits.2 Place a decimal point between these two digits. This is the number between 1 and 10.3 Count the digits between the new and old decimal point. This is the power of 10.4 Power of 10 is positive for larger numbers and negative for small numbers.

Example 4: Expressing a number in standard form 4C

The land surface of the Earth is approximately 153 400 000 square kilometres. Express this area more conveniently by using scientific notation.

SOLUTION:

1 The first two non-zero digits are 1 and 5.2 Place the decimal point between these numbers.3 Count the digits from the old decimal point (end of the

number) to the position of the new decimal point.4 Large number indicates the power of 10 is positive.5 Write in standard form.

1.534

1.53 400 000 – eight digits

Power of 10 is 8+ or 8153400000 1.534 108= ×


Chapter 4 Measurement and energy138 4C

Example 5: Writing numbers to significant figures 4C

Write these numbers correct to the number of significant figures indicated.a 153400 000 (three significant figures)b 0.000657 (two significant figures)

SOLUTION:

1 Write in standard form.2 Count the digits in the number.3 Round to three significant figures.4 Write the answer in standard form correct

to three significant figures.5 Write in standard form.6 Count the digits in the number.7 Round to two significant figures.8 Write the answer in standard form correct

to two significant figures.

a = ×153 400 000 1.534 108

1.534 has 4 digits1.53 rounded to 3 sig. fig.

= ×153 400 000 1.53 108

b = × -0.000 657 6.57 10 4

6.57 has 3 digits6.6 rounded to 2 sig. fig.

= × -0.000 657 6.6 10 4

Significant figuresSignificant figures are used to specify the accuracy of a number. They are often used to round a number. Significant figures are the digits that carry meaning and contribute to the accuracy of the number. This includes all the digits except the zeros at the start of a number and zeros at the end of a number without a decimal point. These zeros are regarded as placeholders and only indicate the size of the number. Consider the following examples.• 51.340 has five significant figures: 5,1, 3, 4 and 0.• 0.00871 has three significant figures: 8, 7 and 1.• 56091 has five significant figures: 5, 6, 0, 9 and 1.The significant figures in a number not containing a decimal point can sometimes be unclear. For example, the number 8000 may be correct to one, two, three or four significant figures. To prevent this problem, the last significant figure of a number can be underlined. For example, the number 8000 has two significant figures. If the digit is not underlined the context of the problem is a guide to the accuracy of the number.

WRITING NUMBERS TO SIGNIFICANT FIGURES

1 Write the number in standard form.2 Count the digits in the number to determine its accuracy (ignore zeros at the end, except

after a decimal point).3 Round the number to the required number of significant figures.



Exercise 4C

1 Write these numbers in standard form.a 7600 b 1 700 000 000c 590 000 d 6 800 000e 35 000 f 310 000 000g 77100 000 h 523 000 000 000i 95 400 000 000 j 540

2 Write these numbers in standard form.a 0.000 56 b 0.000 068 7c 0.000 000 812 d 0.0043e 0.000 058 f 0.000 00312g 0.26 h 0.092i 0.000 000 000 167 j 0.000 06

3 A microsecond is one millionth of a second. Write 5 microseconds in standard form.

4 Sharks existed 410 million years ago.a Write this number in standard form.

b Express this number correct to one significant figure.

5 Write each of the following as a basic numeral.a ×1.12 105 b ×5.34 108

c ×5.2 103 d ×8.678 107

e ×2.4 102 f ×7.8 109

g ×3.9 106 h ×2.8 101

i ×6.4 104 j ×3.5 104

6 Write each of the following as a basic numeral.a × -3.5 10 4 b × -7.9 10 6

c × -1.63 10 7 d × -5.81 10 3

e × -4.9 10 2 f × -9.8 10 1

g × -4.12 10 8 h × -6.33 10 5

i × -3.0 10 9 j × -7.134 10 9

LEVEL 1

Example 4


Chapter 4 Measurement and energy140 4C

7 Convert a measurement of × -5.81 10 3 grams into kilograms. Express your answer in standard form.

8 Evaluate the following and express your answer in standard form.a × × ×(2.5 10 ) (5.9 10 )3 6

b × × ×(4.7 10 ) (6.3 10 )5 2

c × × ×- -(7.1 10 ) (4.2 10 )5 2

d × × ×- -(3.0 10 ) (6.2 10 )4 5

9 Evaluate the following and express your answer in standard form.

a ×× -

9.1 102.8 10

5

2

b ×× -

7.2 104.8 10

7

3

c ××

-

-4.8 103.2 10

4

5

10 Write these numbers correct to the number of significant figures indicated.a 1561 231 (2 sig. fig.) b 3677 720 (4 sig. fig.) c 789 001 (5 sig. fig.)

d 3 300 000 (1 sig. fig.) e 777 777 (3 sig. fig.) f 3194 729 (5 sig. fig.)

g 821 076 (4 sig. fig.) h 7091 (1 sig. fig.) i 49172 (2 sig. fig.)

11 Write these numbers correct to the number of significant figures indicated.a 0.0035 (1 sig. fig.) b 0.191 785 (4 sig. fig.) c 0.001592 (3 sig. fig.)

d 0.111 222 33 (6 sig. fig.) e 0.000 0271 (1 sig. fig.) f 0.019 832 6 (5 sig. fig.)

g 0.00812 (2 sig. fig.) h 0.092 71 (3 sig. fig.) i 0.000 419 (2 sig. fig.)

12 A bacterium has a radius of 0.000 015765m. Express this length correct to two significant figures.

13 Convert a measurement of 2654 kilograms into centigrams. Express your answer correct to two significant figures.

14 Convert a measurement of 4 239 810 milligrams into grams. Express your answer correct to four significant figures.

Example 5



LEVEL 2

15 If y x= 12

2, find the value of y when:

a x = ×2.4 103 b x = × -9.8 10 3

16 The arc length of a circle is θ π= °l r360

2 where θ is the angle at the centre and r is the

radius of the circle. Use this formula to calculate the arc length of a circle when θ = °30 and r = ×7.4 108. Answer in standard form correct to one significant figure.

17 Given that V rh

= find the value of r in standard form when:

a V = ×5 104 and h = ×9 106 b V = × -6 10 7 and h = ×4 102

18 Use the formula E md= 2 to find d correct to three significant figures, given that:a =m 0.08 and E = ×5.5 109 b m = ×2.7 103 and E = ×1.6 104

19 Find x, given x = ×2.7 103 12. Answer correct to four significant figures.

20 Light travels at 300000 kilometres per second. Convert this measure to metres per second and express this speed in standard form.

LEVEL 3

21 Use the formula p3E q= - to evaluate E, given that p = ×7.5 105 and q = ×2.5 104. Answer in standard form correct to one significant figure.

22 The volume of a cylinder is hπ=V r2 where r is the radius and h is the height of the cylinder. Use this formula to calculate the volume of the cylinder when r = ×5.6 104 and h = ×2.8 103. Answer in standard form correct to three significant figures.

23 The Earth is ×1.496 108 km from the Sun. Calculate the distance travelled by the Earth in a year, using the formula 2C rπ= . Answer in standard form correct to two significant figures.


Chapter 4 Measurement and energy142 4D

Food and nutrition

Food provides our bodies with energy, and nutrients for growth and repair. Food energy is a form of chemical energy, and is measured in kilojoules. A kilojoule is 1000 joules. The common unit for food energy used to be the ‘calorie’ (Cal), but the SI unit ‘kilojoule’ (kJ) is now used internationally (1 calorie 4.184 kilojoules)= . If you consume a lot of food with a high kilojoule rating you may be getting more energy than you need. The excess energy is stored as fat.

The healthy eating pyramid on the right shows how food is placed into groups. It suggests the amount of each food category that a person should eat each day, with the more of the food groups at the bottom of the pyramid being eaten than the foods at the top.

4D

FOOD ENERGY

Food energy is measured in kilojoules (kJ). =(1calorie 4.184 kilojoules)

Example 6: Using units of energy related to food and nutrition 4D

The number of kilojoules your body requires each day depends on your age, gender and life style. Answer the following questions using the table.a Jenny is 18 years old. How many kilojoules does she need each day?b Mitchell is 25 years old and works out at the gym for 2 hours. A gym workout uses

2500 kJ per hour. How many kilojoules does he need each day?

SOLUTION:

1 Read table using women aged 18 35- .2 Calculate the energy used in the gym workout.

Multiply 2500 (per hour) by 2(2h)3 Add the energy used in the gym workout to the

daily kilojoules needed for a man aged 18 35- .

a Jenny needs 9000 kJb Gym workout 2 2500 5000 kJ= × =

= +=

Energy 12 500 500017 500 kJ

Age Men kJ per day Women kJ per day

18 35- 12 500 9000

36 70- 10 000 8000


143 4D Food and nutrition

Food labelsFood manufactures are required to label the energy content of their products, to help consumers control their energy intake. The energy available from the food is usually given on labels for 100 g, for a typical serving size and/or the entire pack contents. The nutrition information panel (eatforhealth.gov.au) below provides a few tips on understanding the food label and shopping for healthy food.

www.eatforhealth.gov.au

Example 7: Using units of energy related to food and nutrition 4D

Use the above food label to answer the following questions.a What is the serving size? b How many kilojoules per 100 g?c What is the amount of fibre per serve? d How much saturated fat per serve?

SOLUTION:

1 The top of the label contains information about the serving size.2 Energy is measured in kilojoules and the per 100 g column is

located on the right-hand side.3 Fibre per serve is highlighted in purple in the label.4 Saturated fat per serve is highlighted in salmon in the label.

a Serving size is 30 g.b 1441kJ per 100 g

c Fibre per serve is 6.4 gd Saturated fat per serve

is 0.1g


http://www.eatforhealth.gov.au

Chapter 4 Measurement and energy144 4D

Exercise 4D

1 Complete (use 1 calorie 4.184 kilojoules= ). Answer to the nearest whole number.a 450 Cal kJ= b 1800 Cal kJ= c 15200 Cal kJ=

d 7600 kJ Cal= e 370 kJ Cal= f 1250 kJ Cal=

2 The table below shows the approximate energy value of some common foods.

Find the number of kilojoules in following breakfast meals.a 3 pieces of toast, butter for toast, 1 serve of cornflakes, milk

b 2 fried eggs, 2 rashers of bacon, 1 glass of orange juice

c 1 serve of cornflakes, 1 banana, milk, 1 glass of orange juice

d 1 fried egg, 1 rasher of bacon, 1 piece of toast, butter for toast

e 1 serve of cornflakes, milk, 1 glass of orange juice, 2 fried eggs

3 The column graph below shows the recommended daily food servings for teenagers.

a How many servings of vegetables are recommended for a girl?

b How many servings of meat are recommended for a boy?

c Which food groups for girls have three servings of food recommended?

d Which food group for boys has four servings of food recommended?

e For which food group is it recommended that boys and girls have the same number of servings?

f Find the difference in the number of servings of grain recommended for girls and for boys.

LEVEL 1

Food kJ Food kJ

Bacon (1 rasher) 960 Milk (small glass) 777

Banana 294 Orange juice (1 glass) 231

Butter (for 1 slice of bread)

105 Potato chips (1 serve) 1512

Cornflakes (1 serve) 441 Toast (1 slice) 420

Egg (fried) 550 Soft drink (1 can) 525

12

8

0

10

4

6

2

VegetableGrain Fruit Milk Meat

GirlsBoys

Example 6


145 4D Food and nutrition

LEVEL 2

4 The food label on the right is provided on a 200 g box of cream biscuits.a How many grams of fat are in the 200 g box?

b What is the energy value of:

i one biscuit?ii five biscuits?iii the box of biscuits?

c What is the energy value of:

i 100 g of biscuits?ii 20 g of biscuits?iii 1g of biscuits?

d How many kilojoules in 4 boxes of biscuits?

e Harrison is on a diet of 7030 kJ. How many biscuits can he consume?

f The recommended dietary intake of protein is approximately 50 g per day. What percentage of the recommended dietary intake comes from eating one biscuit?

g The recommended dietary intake of sodium is approximately 1600 mg per day. What percentage of the recommended dietary intake comes from eating one biscuit?

h How many biscuits need to be eaten to provide 90 g of protein?

i What percentage of the total carbohydrate in each biscuit is from sugars?

j What percentage of each box of biscuits is fat?

LEVEL 3

5 The table below shows the recommended daily food servings.

a How many servings are recommended for each day?

b Construct a sector graph to represent this data.

c What percentage of the recommended daily food servings is grain? Answer to the nearest whole number.

d What percentage of the recommended daily food servings is fruit and vegetables? Answer to the nearest whole number.

e Cindy needs 8700 kilojoules of energy in her diet. How many kilojoules should she get from grain? Assume that every serving has the same number of kilojoules. Answer to the nearest whole number.

f William needs 10500 kilojoules of energy in his diet. How many kilojoules should he get from fruit and vegetables? Assume that every serving has the same number of kilojoules. Answer to the nearest whole number.

Grain Vegetable Fruit Milk Meat

9 4 3 3 2

per biscuit per 100gEnergy 95kJ 2090 kJ

Protein 0.5g 11.0 g

Fat, total – saturated

1.2g0.6g

26.4 g13.2 g

Carbohydrate, total– sugars

2.4 g0.4 g

52.8g8.8g

Fibre 0.3g 6.6g

Sodium 25mg 550 mg

Potassium 10 mg 220 mg

Example 7


Chapter 4 Measurement and energy146 4E

Name Symbol Meaning Value Example

Milliwatt mW One thousandth of a watt -10 W3 Small laser pointer

Watt W One watt 10 W0 Smartphone making a call

Kilowatt kW Thousand watts 10 W3 Electric heater

Megawatt MW Million watts 10 W6 Large diesel generator

Gigawatt GW Billion watts 10 W9 Very large power station

Terawatt TW Trillion watts 10 W12 Worldwide nuclear power

Energy consumption

Energy is the capacity to do work. Energy exists in numerous forms, such as heat. The joule (symbol J) is the unit of energy used by the International System of Units (SI). Heat energy, such as that produced by burning natural gas in the home, is usually measured in megajoules (one million joules), symbol MJ.

PowerPower is the rate at which energy is generated or consumed. The watt is the SI unit of power and is equal to one joule per second. The symbol for the watt is W. As for the joule, the standard SI prefixes such as milli, kilo, mega, giga and tera are then added and commonly used to measure power.

4E

Electrical energyThe joule is not a practical unit for measuring electrical energy in most settings. It is more helpful to think of electrical energy in terms of the power drawn by an electrical device, and the length of time the device is in use. For example, when a device with a power rating of 100 W is turned on for one hour, the amount of energy used is 100 watt-hours (Wh). This is the same amount of energy a 50 W device would use in 2 hours.

The kilowatt-hour (kWh) is commonly used to measure electrical energy in household electricity meters. It represents the amount of electrical energy when a 1000 W power load is drawn for one hour. It is the result of multiplying power in kilowatts and time in hours. Note the conversion

=1kWh 3.6 MJ.

Consumption is a rate expressed as an amount over time. Electrical energy consumption can be expressed in units of energy (such as kilowatt-hours or megawatt-hours, or megajoules) consumed per unit of time. The average annual energy use per household in Australia is 11MWh/y or 3MJ/y, which also equates to about eight tonnes of CO2 emissions.


147 4E Energy consumption

Energy rating of appliancesIn Australia an energy rating label is provided for various appliances. It allows consumers to compare the energy efficiency of similar products. Energy rating labels all have a simple star rating. The more stars on the label, the more energy efficient the appliance. The energy consumption figure is the number in the red box. It indicates the amount of electricity (kWh) the appliance typically uses in a year. The lower the number the less the appliance will cost to run. The running cost of the appliance is calculated by multiplying the energy consumption figure by the electricity price rate. Electricity suppliers usually give prices per kilowatt-hour.

ENERGY CONSUMPTION

Energy consumption is the amount of energy consumed per unit of time.

Example 8: Calculating the cost of running appliances 4E

Determine the cost of running the following appliances given the average rate for electricity is $0.17 per kWh.a A dishwasher with an energy consumption of 670 kWh per yearb A 2.4 kW fan heater for five hoursc A 20 W LED light bulb for a year

SOLUTION:

1 Determine the cost by multiplying the energy consumption figure by the electricity price rate.

2 Determine the energy (kWh) by multiplying the power rating (kW) by the hours used.

3 Determine the cost by multiplying the energy by the electricity price rate.

4 Determine the energy (kWh) by multiplying the power rating (kW) by the hours used.

5 Determine the cost by multiplying the energy by the electricity price rate.

a = ×=

Cost 670 0.17$113.90

b Electricity 2.4 5

12 kWh

= ×=

= ×=

Cost 12 0.17$2.04

c Electricity 0.02 24 365

175.2 kWh

= × ×=

Cost 175.2 0.17

$29.78

= ×=



Exercise 4E

1 Answer the following questions using this electricity account.

a When was this electricity account issued?

b What is the total amount due?

c What is the due date?

d How much GST was charged?

e What is the charge for the peak use of electricity?

f How much has the bill increased from the last bill?

g Express the increase as a percentage of the last bill. Answer correct to two decimal places.

LEVEL 1


149 4E Energy consumption

2 A washing machine has an energy consumption figure of 206 kWh per year. Calculate the cost of running the washing machine for a year using the following electricity price rates.a $0.2185 / kWh

b $0.2419 / kWh

c $0.2653 / kWh

3 Leo uses a 800 W microwave oven for a total of 20 hours per week.a How much energy does the microwave oven use

per week?

b What is the cost of using the microwave oven for a week if the electricity is charged at a rate of $0.2367 per kWh?

4 The Bailey family has a television set with a 200 W rating.a How many kilowatt-hours of electricity does it use

in a week if it is switched on for an average of 9 hours per day?

b What is the cost of running the TV for a week if electricity is $0.2259 per kWh?

5 Samantha uses a hairdryer with a rating of 1.5kW for a total of 8 hours. Her electricity is charged at a rate of 22.81 cents per kWh.a How many kilowatt-hours were used by the hairdryer?

b What is the cost of using the hairdryer? Answer to the nearest cent.

6 Darcy uses an iron with a rating of 500 W for an average of 8 hours per week. Find the cost of the electricity used to do the weekly ironing if it is charged at a rate of 25.67 cents per kWh.

7 Bian received her natural gas account.

a Calculate the cost of the first 5500 MJ. Answer to the nearest cent.

b Calculate the cost of the next 13900 MJ. Answer to the nearest cent.

c What is the total charge? Answer to the nearest cent.

Amount used Charges Cost

First 5500 MJ 2.491 cents per MJ

Next 13900 MJ 1.499 cents per MJ

Supply fee $40.00

Example 8



LEVEL 2

8 The power ratings of various electrical appliances are shown in the table.

a How much energy is used by the television for 5 hours? Answer in kilowatt-hours.

b How much energy is used by the toaster for 30 minutes? Answer in watt-hours.

c For how many hours was the dishwasher used if its cost of operation was $11.76? The cost of electricity is $0.2450 per kilowatt-hour.

d For how many hours was the hair dryer used if its cost of operation was $1.95? The cost of electricity is 26 cents per kilowatt-hour.

LEVEL 3

9 An energy company’s charges for gas over a 3-month period are shown in the table.

a Paige used 3580 MJ of gas in this period. What is the cost of this gas? Answer to the nearest cent.

b Muhammad used 4250 MJ of gas in this period. What is the cost of this gas? Answer to the nearest cent.

c What percentage of Muhammad’s gas usage was charged at the lower rate? Answer correct to one decimal place.

d Jesse received a gas bill for $93.24. How much gas did he use in this period?

e Ellie has decided to reduce her energy bills. She has a target of $50 for gas. What is the maximum number of MJ she can use in this period?

f Aaron used 5620 MJ of gas in this period. The gas charges are increasing by 5% in next quarter. However, Aaron has purchased a new gas heater with a 5-star rating that will reduce his consumption by 800 MJ. Calculate Aaron’s expected bill next quarter.

Electrical appliance Power rating

Dishwasher 1600 W

Hair dryer 600 W

Light bulb 75W

Refrigerator 150 W

Television 300 W

Toaster 1000 W

Usage chargeFirst 4000 MJ $0.02590 per MJ

Additional MJ over 4000 $0.01573 per MJ


151

Sum

mar

y

Chapter 4 Summary

Key ideas and chapter summary

Units of measurement

mega

kilo

unit

centi

milli

÷ 1000

÷ 1000

÷ 100

÷ 10

× 1000

× 1000

× 100

× 10

10 000 cm 1 m

1 ha 10 000 m

1000 000 cm 1 m

2 2

2

3 3

==

=

Measurement errors

Precision or limit of reading - smallest unit on the measuring instrumentAbsolute error the difference between actual value and measured value,

or 12

precision

-

± ×

Upper bound = Measurement + Absolute error

Lower bound = Measurement - Absolute error

Percentageerror Absolute errorMeasurement

100%( )= ± ×

Writing numbers in standard form

1 Find the first two non-zero digits.

2 Place a decimal point between these two digits.

3 Power of 10 is number of the digits between the new and the old decimal point. (Small number - negative value, large number - positive value)

Food and nutrition

Food energy is measured in kilojoules. If you consume a lot of food with a high kilojoule rating you may be getting more energy than you need. The excess energy is stored as fat.

) )( (=1calorie Cal 4.184 kilojoules kJ .

The energy available from the food is usually given on labels for 100 g, for a typical serving size and/or the entire pack contents.

Energy consumption

Energy is the capacity to do work. The joule (J) is the basic SI unit of energy.

Power is the rate of consuming or generating energy. Its SI unit is the watt (W) which is equal to one joule per second.

Electricity meters measure electrical energy in kilowatt-hours, kWh, which is the energy consumed by running a 1 kW appliance for one hour. 1 kWh 3.6 MJ=

Running cost of the appliance is calculated by multiplying the energy consumption by the electricity price in cents per kilowatt-hour.


152Re

view

Chapter 4 Measurement and energy

Multiple-choice questions

1 Convert 7.5 metres to millimetres

A 0.0075 mm B 75 mm C 750 mm D 7500 mm

2 Convert 0.85 g to kilograms.

A 0.00085 kg B 0.0085 kg C 85 kg D 850 kg

3 How many square millimetres are in a square centimetre?

A 10 B 100 C 1000 D 10 000

4 6.2 m2 is the same as:

A 620 cm2 B 0.0062 km2 C 62000 cm2 D 6200 mm2

5 What is the absolute error if the precision is 0.1 cm?

A 0.005 cm B 0.05 cm C 0.5 cm D 5 cm

6 Write 4500 000 in standard form.

A 4.5 10 6× - B 4.5 10 5× - C 4.5 105× D 4.5 106×

7 Express 0.0655 correct to two significant figures.

A 0.06 B 0.07 C 0.065 D 0.066

8 What is the internationally accepted unit to measure food energy?

A Calorie B Gram C Kilowatt D Kilojoule

9 Flynn uses a 1.2-kilowatt dishwasher for a total of 5 hours. He is charged at a rate of 25.72 cents per kilowatt-hour. What is the cost of using the dishwasher?

A $0.31 B $1.29 C $1.54 D $6.00

10 How much energy does a 600-watt hair dryer use running for 5 hours?

A 1.2 kWh B 3 kWh C 120 kWh D 3000 kWh


153

Revi

ew

Chapter 4 Review

Short-answer questions

1 There are six tonnes of iron ore on a train. What is the mass (in tonnes) if another 246 kg of iron ore is added to the train?

2 Complete the following

a =500 cm m2 2 b 4000 cm = mm2 2 c 3km = m2 2

3 A field has a perimeter of exactly 400 m. Lily measured the field to be 401.2 m, using a long tape marked in 0.1 m intervals.

a Calculate the limit of reading.

b What is the absolute error for Lily’s measurement?

c What is the percentage error for Lily’s measurement? Answer correct to three decimal places.

4 Write as a basic numeral.

a 4.8 106× b 6.25 10 4× - c 35 124×

5 Write these numbers in standard form

a 50 800 b 0.0036 c 381 000 000

6 Evaluate the following and express your answer in standard form.

a (7.2 10 ) (2.1 10 )5 4× × × b 4.6 102.3 10

4

2×× - c 5.8 10

2.9 10(3.5 10 )

5

46×

×× ×

-

7 Convert a measurement of 3580 tonnes into milligrams. Express your answer in standard form correct to two significant figures.

8 Find the value of 45 154× and express your answer in standard form correct to two significant figures.

9 The table below shows foods and the energy in kilojoules in a normal size serving.

Find the number of kilojoules in following meals.a Fish and chips, ice cream and soft drinkb Roast lamb, chips, apple pie

c Ice cream, apple pie, soft drink

Food KJ Food KJ

Chips 1425 Apple pie 1380

Fish 340 Ice cream 810

Roast lamb 1064 Soft drink 372


154Re

view

Chapter 4 Measurement and energy

10 The graph on the right is part of an electricity account issued to a customer.

a How many times per year is the electricity meter read?

b What unit is used to measure electrical energy?

c What is the energy consumption in the January quarter?

d What is the energy consumption in the July quarter?

e Which quarter used 6 kWh of electricity?

f Which quarter used 9 kWh of electricity?

g Which quarter had the greatest usage of electricity?

Extended-response question

11 The power rating for electrical appliances is shown in the table.

a How much energy does a laptop computer use for 12 hours?

b How much energy does a microwave use for 12

hour? Answer in watt-hours.

c For how many hours was the vacuum cleaner used, if the cost of operating the vacuum cleaner was $6.36? The price of electricity is $0.2650 per kilowatt-hour.

2

4

6

8

10

12

Apr0 Jul Oct Jan Apr

kWh

Electrical appliance Power rating

Electric shaver 15 W

Hair dryer 1000 W

Iron 900 W

Laptop computer 50 W

Microwave 1200 W

Vacuum cleaner 600 W


Measurement and energy - web2.hunterspt-h.schools.nsw.edu.auweb2.hunterspt-h.schools.nsw.edu.au/studentshared/MATHEMATICS… · Chapter 4 . Measurement and energy. 4A. Units of measurement.

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