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2339Unit 661MathematicsGreek Letters

In addition to Roman letters, there are a number of Greek letters used in Engineering, such as follows:

α alpha mathematical symbol for "is directly proportional to" temperature coefficient of resistance coefficient of linear expansion

β beta

γ gamma

Δ delta mathematical symbol for "change"

ε epsilon permittivity

ζ zeta

η eta efficiency

θ theta often used to denote an unknown angle

ι iota

κ kappa

λ lambda wavelength

μ mu permeability

ν nu

ξ xi

ο omicron

π pi ratio of circumference to diameter

ρ rho density;resistivity

Σ sigma mathematical symbol for "sum of"

τ tau time constant

υ upsilon

φ phi magnetic flux

χ chi

ψ psi electric flux

Ω omega Ohms

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What is BODMAS?

Is an an Acronym (word for letters used to shorten a collection of words)

BODMAS enables us to know exactly the right sequence of doing things mathematically.

In particular electronic calculators have to use a rule (known in computing circles as an algorithm) to know which answer to calculate when given a string of numbers to add, subtract, multiply, divide etc.

What do you think the answer to 2 + 3 x 5 is?

Is it (2 + 3) x 5 = 5 x 5 = 25?

or 2 + (3 x 5) = 2 + 15 = 17?

BODMAS can come to the rescue and give us rules to follow so that we always get the right answer:

(B)rackets(O)rder(D)ivision(M)ultiplication(A)ddition(S)ubtraction

According to BODMAS, multiplication should always be done before addition, therefore 17 is actually the correct answer according to BODMAS and will also be the answer which your calculator will give if you type in 2 + 3 x 5 <enter>.

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We should know from school what everything in BODMAS means apart from "Order". Order is actually a poor word to use here "Power" would be much better though BPDMAS doesn't quite have the same ring to it!

Order means anything raised to the power of a number.

You may have heard of Einstein's famous equation E = mc2 here it can be said that c is raised to the power 2, or c has order 2 or c is squared (they all mean the same thing!).

Here's an example to show how to use all the features of BODMAS:

Explain the answer that a calculator would give to the calculation 4 + 70/10 x (1 + 2)2 - 1 according to the BODMAS rules.

Brackets gives 4 + 70/10 x (3)2 - 1

Order gives 4 + 70/10 x 9 - 1

Division gives 4 + 7 x 9 - 1

Multiplication gives 4 + 63 - 1

Addition gives 67 - 1

Subtraction gives 66

The Part of BODMAS Most Often ForgottenIt is quite common for people to forget that brackets always come first - which means you can calculate the answer of whatever is in the brackets before you attempt to calculate the rest of the problem.

Once you've worked out everything in the brackets, normally these sort of problems become very easy!

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The Theorem of Pythagoras

Pythagoras was a Greek who lived over 2000 years ago.

He didn't actually invent the theorem, which bears his name, as the result had been known for hundreds of years before his time, what he did was to prove that the rule worked always, and not just for particular sets of numbers.

The rule states that, in a right-angled triangle, the square on the hypotenuse equal to the sum of the squares on the other 2 sides.

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The longest side is called the hypotenuse

Square A

Square B

Square C

Pythagoras Theorem also states the

Square A = Square B + Square C

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Example 1 Work out the area of Square A

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Square A?? m2

Square B5m2

Square C32 m2

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Square A?? m2

Square B4.3m2

Square C10.1 m2

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Square A10.1 m2

Square B4.3m2

Square C?? m2

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Example 1 Work out the area of Square A Your answer should be:

5 + 32 = 37 m2

Example 2 Work out the area of Square A Your answer should be:

4.3 + 10.1 = 14.4 m2

Example 3 Work out the area of Square C Your answer should be:

If Square A = Square B + Square C

Then

Square C = Square A – Square C

Square C = 10.1 – 4.3 = 5.8 m2

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Work out the following Questions

1.

a) 18 cm2

b) 324 cm2

c) 234 cm2

2.

a) 32 cm2

b) 32m2

c) 30 m2

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Square A3CM2

15CM2

Square A4M2

28M2

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3.

a) 7.2m2

b) 11.8m2

c) 12m2

4.

a) 1.2cm2

b)13.8 cm2

c) 13.8m2

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Square A

2.3M2

9.5M2

7.5cm2

6.3CM2

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4.

5.

a) 4.5 mb) 45cm c) 4.5 m2

6.

a) 4.87 m2

b)6.57 m2

c) 43.16 m2

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19.2m2

14.7M2

5.72m2

0.85M2

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Question

1 A

2 B3 B4 A

5 C

6 A

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Edge Calculation

A square has four sides, or you could say four edges.

Obviously, a square has equal sides. So, if you know the length of an edge you can find the area by multiplying the edge by itself

An example:

The edge of the square is 4.7cm

The area is therefore: 4.7 * 4.7 = 22.09 cm2

On your calculator you could have typed in 4.7 and pressed the X2 your display will then change to the answer 22.09

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Find the areas of these squares

1. Edge 8.3 cm

a) 16.6 cm2b) 68.89 cm2c) 24 cm2

2. Edge 12.5 m

a) 25 m2b) 25 mc) 156.25 m2

3. Edge

a) 18.4 cm2b) 9.2 cm2c) 21.16 cm2

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4.6 cm

Question

1 B

2 C3 C

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If you know the area, you can find the length by using the Square root key on the calculator

√ x

An example:

Area of a Square = 59.7 cm2

For length, entering into the calculator √59.7will give the answer 7.72657750883

Which is 7.73 to 2 d.p.

An example:

Area of a Square = 49 cm2

For length, entering into the calculator √49will give the answer 7

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Examples:

Find the length, or edge, of these squares:

1. Area = 100 m2, so what is the edge length?

a) 10 m

b) 10000m

c) 100m

2. Area = 625 cm2, so what is the edge length?

a) 312.5 mb) 25 cmc) 156.25 cm

3. Area = 587.2 m2, so what is the edge length?

a) 146.8 mb) 293.6 cmc) 24.23 m2

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Trigonometry

We will now combine the results of the previous exercises to find one side of a right-angled triangle, given the other 2 sides.

We have discussed that the hypotenuse = B2 + A2

Side B is 3 *3 or 32 = 9 cm2

Side A is 4 * 4 or 42 = 16 cm2

So Square C = 9 + 16 = 25 cm2

So √25 = 5 cm

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3 cmSide B

4 cm Side A

Hypotenuse Side C

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Examples

1. Calculate the hypotenuse edge length.

So: a2 + b2 = c2

Or Opposite2 + adjacent2 = Hypotenuse2

6.32 + 3.22 = c2

39.69 + 10.24 = 49.93√49.93=7.06611633077=7.07¿2d . p .

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3.2 mSide B

6.3 m Side A

Hypotenuse Side C

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2. A square edge is 36cm long, so calculate the diagonal length indicated in red.

So: a2 + b2 = c2

Or Opposite2 + adjacent2 = Hypotenuse2

362 + 362 = c2

1296 + 1296 = 2592√2592=50.9116882454=50.91 ¿2d . p .

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Example: Find the length of the hypotenuse:

1.

a) 14 m

b) 10 m

c) 28 m

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6 mSide B

8 m Side A

Hypotenuse Side C

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2.

a) 17 m

b) 34 m

c) 13 m

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5 cmSide B

12 cm Side A

Hypotenuse Side C

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3.

a) 9 cm

b) 12.7 cm

c) 81 cm

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5.9 cmSide B

6.8 cm Side A

Hypotenuse Side C

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4.

a) 16.1 mm

b) 80.36 mm

c) 112.5 mm

d)

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48.2 mmSide B

64.3 mm Side A

Hypotenuse Side C

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5.

a) 31.74 m

b) 44.5 m

c) 63.8 m

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19.3 mSide B

25.2 m Side A

Hypotenuse Side C

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6.

a) 5 km

b) 50 km

c) 62 km

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14 kmSide B

48 km Side A

Hypotenuse Side C

2339Answers

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Question

1 B

2 C3 A4 B5 A6 B

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Example: Find the missing side:

1.

Note that in this question, we know the length of the hypotenuse. The missing side is one of the shorter sides of the triangle.

Now, since small square + medium square = large square

it follows that

medium square = large square − small square

So we use this version of Pythagoras:

x2 = (6·5)2 - (2·5)2 Note the minus sign instead of the plus sign

= 42·25 - 6·25

= 36

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2.5 cm

x

6.5 cm

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Find the lengths of the missing sides:

1.

a) 1.8m

b) 0.6m

c) 5.4m

2.

a) 28mm

b) 56mm

c) 112mm

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3.0 m

2.4 m

X

70mm

42mm

X

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3.

a) 8.4m

b) 12.6m

c) 2.6m

4.

a) 4.3cm

b) 12.9cm

c) 7.45cm

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9.1m

3.5 m

X

8.6cm

4.3cm

X

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5.

a) 60mm

b) 278mm

c) 130mm

6.

a) 77.25cm

b) 0.992m

c) 9m

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132mm

145mm

X

1.25m

76cm

X

Question

1 A

2 B3 A4 C5 A6 B

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Exercise 6

In this exercise, the missing sides are all mixed up. Sometimes it is the hypotenuse you want to find, sometimes it is one of the other sides.

If there is no sketch, make sure you draw one!

In a couple of the questions you are led through an example before tackling one on your own.

1)

2)

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5.7cm

3.9cm

X

142mm

61mm

X

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3. In this question you are to find the length of the line CD.

Step 1: Look at the small triangle ABC. Find the length of BC.Step 2: Look at the large triangle ABD. Find the length of BD.Step 3: What's the length of CD then?

4. Follow the procedure of the previous question to find the length of XY.

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20m12m

15m

A

BC D

12.6cm7.2cm

8.7cm

W

X YZ

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5. The picture shows the side view, not to scale, of a 20 m long swimming pool which slopes gradually from 1 m down to 3 m.

Find the length of the sloping edge.

Step 1 Split the shape into a rectangle and a triangle.

Step 2 Insert the distances in the triangle diagram.

Step 3 Use Pythagoras in the triangle to find the hypotenuse.

6. Use the procedure from the previous question to calculate the length of the slope in this diagram.

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3m

1m

20m

25m

3.5m0.9m

10m

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7. PQ is a 95 m high TV mast. XY is one of the wires supporting it. The wire is 75 m long. The wire is fixed to the ground 40 m from the foot of the mast.

a. How far up the mast does the wire reach?

b. How far from the top of the mast is this?

8. Jim is building the foundations of a greenhouse. He isn't sure if the corner is an exact right angle or not, so he takes the measurements shown in the diagram. Are his foundations OK or not?

9. Find the perimeter of this triangle. (Remember, perimeter = sum of all the edges.)

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P

X

Q Y

4.9M

3.4M

6.1M

53MM

87MM

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10.The picture shows a lean-to bike shed.

a) Find the length of the sloping edge XY.

b) Find the area of the sloping roof and the cost of roofing felt if 1 m2 costs £2.49.

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3.1M

2.8M

2.6M

10.4M

X

Y

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Click here to check your answer for exercise

1 6.91cm

2 128.2mm

3 7m

4 5.46cm

5 20.1m

6 15.22m

7a 63.4m

7b 31.6m

8 No. Diagonal should be 5.96

9 241.9mm

10a 2.84m

10b £73.53

1.

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Ratios

Exercises

1. Sam and her younger brother share sweets in the ratio of 4:1. Who gets more sweets, Tania or her brother?

2. George spends 10 times as much on accommodation as on transport. Complete this sentence. George spends money on accommodation and transport in the ratio of ……… :………..

3. The ratio of cars to motorcycles on the roads is 25:1. Explain what this means in a sentence.

4. The ratio of trolleys to handbaskets at Home Bargains is 1:1. What is another way of saying this?

5. Write down the ratio of triangles to squares:

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Question

1 Sam

2 10:13 The number of cars is 25 times the number

of motorcycles4 The same amount of trolleys is at Home

Bargains as there are hand baskets5 2:1

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Map Scales

Exercises

1) Here are 2 maps showing Perth. One a scale of 1: 1 Mile, the other 1: 5 Mile. Which is which?

Map A

Map B

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2) Work out how many cm this river measures in real life, then convert your answer to m then convert that answer to km.

A scale 1:250000 and a map measurement is 8 cm.

3) How many km would this road measure in real life?

A scale of 1:3000 and a map measurement of 16cm

4) How many cm would this journey appear as on a map with scale 1:2 000 000?M6 Birmingham 250km

5) Work out how many cm this journey would appear as on a map withscale 1:250 000.A9 Inverness 40km

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Check your answer

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Question

1 Map A1: 1 Mile Map B 1: 5 Mile

2 20km3 0.48km4 12.5 cm5 16 cm

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Exercise 9

1 The ratio of tables to chairs in a room is 1:4a If there are 7 tables in the room, how many chairs are there?b If there are 20 chairs in the room, how many tables are there? 2 The ratio of males to females in a class is 2:1a If there are 18 males in the class, how many females are there?b If there are 6 females in the class, how many males are there? 3 At a campsite the ratio of caravans to tents is 5:1a If there are 10 tents at the campsite, how many caravans are there?b If there are 10 caravans at the campsite, how many tents are there?

4 The ratio of vodka to orange in a drink is 1:3 a If there are 180 mls of vodka in the drink, how much orange is there?b If there are 300 mls of orange in the drink, how much vodka is there? 5 The ratio of chairs to tables in a bar was 5:2a If there were 20 chairs, how many tables were there?b If there were 18 tables, how many chairs would there be? 6 At a football match, the ratio of City fans to Rovers fans was 3:8a If there were 1200 City fans, how many Rovers fans were there?b If there were 400 Rovers fans, how many City fans were there?

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Answer1a 1:4 = 7:281b 1:4 = 5:20 2a 2:1 = 18:92b 2:1 = 12:6 3a 5:1 = 50:103b 5:1 = 10:2 4a 1:3 = 180:5404b 1:3 = 100:300 5a 5:2 = 20:85b 5:2 = 45:18 6a 3:8 = 1200:32006b 3:8 = 150:400

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Ratios Expressed as Fractions

A ratio of (say) 1:2 is the same ratio as 2:4,

3:6, 4:8 etc. A ratio of (say) 2:3 is the same

ratio as 4:6, 6:9, 8:12 etc.

We notice that in the above examples we could have written:

12 is the same fraction as

24 ,

36, 48

23 is the same fraction as

46 , 69, 812

In Physics and Engineering, when one quantity is divided by another quantity we often speak of the resulting quantity as a ratio. This means ratio is expressed in fraction form.

e.g. resistance = voltage divided

by current or R = VI

the VI part of the above expression is called the ratio of voltage divided by current.

It could be written as V: I but, in practice, it never is!

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A number which can be expressed as a fraction or ratio of 2 whole numbers is called a rational number.

A number which cannot be expressed as the ratio of 2 whole numbers is called an irrational number. Examples of irrational numbers are π and √2

Proportion

In the above examples we had:

12=24=¿ 36=

48 and 23=¿ 46 = 69=

812

If we do the sums and divide the top lines of the fractions by the bottom lines we get:

0.5 = 0.5 = 0.5 = 0.5 and 0.67 = 0.67 = 0.67 = 0.67

In each example the answer is the same number. That is to say, the

answer is a constant. Again if we take the example of VI

V is said to directly proportional to I.

This is written as V α I

where α is the symbol for "is directly proportional to"

If V α I, we can write it as V = constant x I

or VI =constant

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Sometimes we find that numbers fit the following pattern

S α Vt

Because the "t" is on the bottom line of the fraction, s is said to be inversely proportional to t.

S = constant x 1t

(In this example, s could be speed, t could be time and the constant would be distance.)

Splitting a Quantity in a Given Ratio

Sometimes we are required to split a quantity into a given ratio.

For example, Adam and Bob are to split £60 in the ratio 1:5.

How much does each get?

Here is how we do it.Add the two numbers of the ratio together. In this example, 1 + 5 = 6.

This becomes the denominator (or bottom line of a fraction.)

Each number in the ratio then becomes the numerator (or top line of a fraction) in turn

Adam would get one sixth of £60 = £10

Bob would get a fifth of £60 = £50To check the answer:

1 Total amount shared = £10 + £50 = £60.

2 Ratio of amounts = £10: £50 = 1:5.

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2339Exercises

1. Dianne and Tracy share an £80 bingo prize in the ratio 1:4. What fraction does each get? How much prize money does each get?

2. Sean and his Dad split the cost of a car costing £1500 in the ratio 3:1. What fraction does each pay and how much money does each contribute?

3. There are 120 calculators in the cupboard consisting of scientific and basic types in the ratio 1:2. What fraction is there of each type? How many are there of each type?

4. A rock group play "cover songs" and original songs in the ratio1:4. If they play 25 songs altogether, how many of each type do they play?

5. A shade of green paint was made by mixing blue paint and yellow paint in the ratio 2:7. If there is 450 ml of the green paint, how much blue and how much yellow were used?

6. In the car park there are cars belonging to staff and belonging to students in the ratio 3:2. If there are 85 cars in the car park altogether, how many cars belong to staff?

7. On a drive through London, Garry met red and green traffic lights in the ratio 5:2. If he encountered 42 sets of lights, how many green lights did he meet?

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Answers:

1. Dianne 1/5 = £16Tracey 4/5 = £64

2. Sean ¾ = £1125Dad ¼ = £375

3. Scientific 1/3 = 40Basic 2/3 = 80

4. 5 cover songs and 20 original

5. 100ml blue and 350ml yellow

6. 51 staff 34 students

7. 30 red 12 green

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2339BODMAS

By convention when a set of mathematical instructions is set together in one line they must be calculated in a definite order.

The order is Brackets, Of, Divide, Multiply, Add, Subtract. ("of" means multiply.)

Also remember that when a number and a letter are placed together with no symbol between them e.g. 2a then it means

2 *a

Similarly, 2 letters together e.g. ab means a * b.

Exercises 11

Calculate:

a) 6+5*2

b) 20+3*3

c) 9+2*2

d) 80 ÷ (5*2)

e) 2/5 of (15 +3)

f) (7+13) * (5-1)

g) 2+5∗24−1

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2339Answers

A 16B 26C 13D 8E 13F 80g 4

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2339Exercises

1. R = d – fCalculate what R comes to when d = 18.4 and f = 2.7

2. C = 5u – 2Calculate C when u = 6

3. y = hj + pCalculate y when h=6, j=44 and p=50

4. The amount of juice left in a bottle after some drinks have been poured is given by a = 1500 – 200d where a is the amount of juice left in millilitres and d is the number of drinks poured. Calculate the amount of juice remaining after 3 drinks have been poured.

5. A child is unwell. The amount of medicine she should take is given by the formula:

where C is the amount of medicine in millilitres and a is the age of the child in years. If the child is 12 years old, how many millilitres of medicine should she take?

6. The time, T minutes, a hillwalker should take for a typical walk is given by

Naismith's Rule: where h is the horizontal distance walked in kilometres and v is the vertical rise in metres. Calculate the number of minutes

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2339needed to do a walk involving 6 horizontal kilometres and incorporating a rise of 800 metres

Answers

1 R = 15.7

2 C = 28

3 Y = 314

4 900 ml

5 2.5ml

6 170 minutes

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Changing the Subject of a Formula

Sometimes we may be given an equation such as:

S = dt

If we know the value of d and the value of t then we can work

out the value of s. But suppose we know s and t and we want to

work out d.

How do we do it?

In problems like this we can often use cross-multiplication.

S = dt is the same as S1 = dt

Multiply the top left by the bottom right, and Multiply the top right by the bottom

left.

We get st = (1)d

This is called cross multiplying. We now have d = s x t

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Another example .

v Pv s

=N p

Ns

(Note: The p and s in the above formula are subscripts. We do not multiply by

them!)

Say we want to rearrange to find Ns

1 Cross multiply. We get VpNs = VsNp

2 "Get rid" of Vp by dividing both sides by Vp

v p Nsv p

=v s Npv p

The Vp cancels giving

vs N p

v p

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Exercise

1. F = ma

a) Find what a equalsb) Find what m equals.

2. P = VI

a) Find what V equals.b) Find what I equals.

3. v2 = u2 + 2as

a. Find what v equals.b. Find what u equals.c. Find what a equals.d. Find what s equals.

4. E = ½mv2

a) Find what m equals.b) Find what v equals.

5. P= v2

R

a) Find what v equals.b) Find what R equals.

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Answers

1a a= Fm

1b m= Fa

2a v=PI

2b pv=I

3a v=√u2+2as 3b U=√v2−2aS

3c a= v2−U 2

2 s3d s= v

2−u2

2a

4a m= E12v2

4b v=√ E1/2m

5a v=√ pR 5b R= v2

P

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Significant Figures

Sometimes we are asked to round numbers to a certain number of significant figures.

Basically, a significant figure is a number other than zero, but, unfortunately, there are exceptions to this rule!

For example, 1234 has 4 figures and also 4 significant figures because none of

the figures is zero.

The same number to 3 significant figures would be 1230.

The same number to 2 significant figures would be 1200.

1.234 has 4 significant figures.

1.2 has 2 significant figures.

0.123 has 3 significant figures. 0.0123 also has 3 significant figures.

But zeroes between nonzero digits are significant e.g. 1.02 has 3 significant figures.

The rules are:

1. All non-zero digits are significant.2. Zeroes between nonzero digits are significant.3. Zeroes to the left of the first non-zero digit are not significant.4. Zeroes to the right after the decimal point are significant. (eg 0.2300 has 4

significant figures. (If it had been meant to be 2 significant figures then the zeroes would have been missed out.)

5. Zeroes to the right-hand side where there is no decimal point may or may not be significant. It depends on the measurement.

To try to explain rule 5.

A stamp collecting crowd may be counted. If there are exactly 4233 spectators, then that is the answer to 4 significant figures. The crowd may be estimated as 4230. That would give the answer to 3 significant figures. The crowd may be estimated as 4200. That would give the answer to 2 significant figures.

But suppose the crowd was exactly 4200. In that case the answer is given to 4 significant figures because the zeroes mean what they say.

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Exercise 14

Round the following to 3 significant figures

a) 5233

b) 5238

c) 523300

d) 0.5233

e) 0.005233

f) 5.0238

g) 5.233

h) 5.2380

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Answers

a 5233 5230

b 5238 5240

c 523300 523000

d 0.5233 0.523

e 0.005233 0.00523

f 5.0238 5.02

g 5.233 5.23

h 5.2380 5.24

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Negative Numbers

Suppose a shopkeeper in the course of his business gains £100 and then loses £70. His net gain is £30.

Suppose another time he gains £70 and then loses £100. His loss is £30.

Another way to express his loss is to say his gain is negative £30 or minus £30 or -£30.

A thermometer may read 20o. If the temperature falls by 30o then the new temperature is -10o.

Heights may be measured from sea level e.g. a hilltop may be 100 metres above sea level.

How would we measure the depth of a fish in the sea?

If it is 10 metres below sea level, then we could say that its height is -10 metres.

We are thus familiar with the idea of negative numbers in various aspects of everyday life.

In algebra we say that numbers are positive if they are preceded by a + sign e.g. +5 (if there is no sign then we assume a +).

Numbers are said to be negative if preceded by a - sign e.g. -5.

Positive and negative numbers can be added together

6 - 3 = 3

3 - 6 = -3

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In engineering problems, it often occurs that brackets have to be removed. The rules

are:

1. When an expression within brackets is preceded by the + sign then the

brackets may be removed without making any change in the expression.

e.g.

a+b-c+(d-e-f) = a+b-c+d-e-f2. When an expression within brackets is preceded by the - sign then the

brackets may be removed if the sign of every term within the brackets is

changed.

e.g.

a+(b-c)-(d-f+e) = a+b-c-d+f-e

Thus

a+(b-c) = a+b-c a-(b+c) = a-b-c a-(b-c) = a-b+c

Also

+(+a) = +a -(+a) = -a +(-a) = -a -(-a) =+a

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Examples

1 27 + (9 – 4)

2 13 – (9 – 4 – 3)

3 9 – (5 – 7 + 1)

4 –5x + (9x – 3x)

5 9x – (5x – 2x)

6 (8a – 5a) – 10a

7 12p – (5p + 2p)

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1 27 + (9 – 4) 32

2 13 – (9 – 4 – 3) 11

3 9 – (5 – 7 + 1) 10

4 –5x + (9x – 3x) x

5 9x – (5x – 2x) 6x

6 (8a – 5a) – 10a -7a

7 12p – (5p + 2p) 5p

Answers

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Surface Area

We have previously looked at the area of a plane surface.

The surface area of a solid is simply the areas of all the faces added together. For example, a cube has 6 square faces. If the sides of the cubes are (say) 0.5 metres then the area of one of the faces is (length times breadth) 0.5 x 0.5 = 0.25 m2.

As there are 6 faces, the surface area of the cube, in this example,

is 6 x 0.25 = 1.5m2.

For solids with flat surfaces we simply add together the areas of each face. Sometimes it is helpful to imagine the surface of the solid made out of folded paper.

When we unfold the paper it leaves a flat piece of paper.

This is called a net.

Below is a net of a cube.

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We have some special formulae for the surface areas of solids with curved surfaces.

Surface Area of a Cube = 6a2

(a is the length of the side of each edge of the cube)

The surface area of a cylinder = 2πrh + 2πr2 where h is the height of the cylinder and r the radius.

The surface area of a cone

= πrs + πr2

where s is the slant height and r the radius. (The slant height is the length of the side as measured, not the vertical height.)

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The surface area of a sphere

is = 4πr2

Volumes

We have previously seen volumes of solid figures with flat faces.

We have some special formulae for the volumes of solids with curved

surfaces.

The volume of a cylinder, = πr2h where h is the height of the cylinder and r the

radius.

The volume of a cone = V=π r2 h3 where h is the vertical height and r the radius.

The volume of a sphere is v=43 π r3

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Exercises

1. What is the surface area of a cylinder of height 0.5

metres and radius 0.1 metres?

a) 0.376 m2b) 0.377 m2c) 0.378 m2

2. If the surface area of a sphere is 5 m2, what is the

radius?

a) 0.632b) 0.631mc) 0.630m

3. What is the volume of a cone of vertical height 0.5 metres and radius 0.1 metres?

a) 0.0052m3b) 0.0051m3c) 0.0050m3

4. If the volume of a sphere is 5 m3, what is the radius?

a) 1.063mb) 1.062mc) 1.061m

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Answers

1 b2 c3 a4 c

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