Tools: For now you will need…. Tools: For now you will need…. PenPen

PaperPaper

CalculatorCalculator

Context…

Mathematics in Construction and the Built environmentMathematics in Construction and the Built environmentConstruction, civil engineering and building services engineering are Construction, civil engineering and building services engineering are technical disciplines which require the collection, processing and use of technical disciplines which require the collection, processing and use of numerical data.numerical data.

Construction team members will include …. Construction team members will include …. Designer – architect – engineer - cost controller - quantity surveyor Designer – architect – engineer - cost controller - quantity surveyor

In complex situations, design engineers use formulae to…. In complex situations, design engineers use formulae to….

calculate bending moments in beams and sizing for structural elements.calculate bending moments in beams and sizing for structural elements.

““It is therefore essential that learners develop an understanding of the It is therefore essential that learners develop an understanding of the mathematical methods and techniques required for key activities, and of mathematical methods and techniques required for key activities, and of how to apply them correctly”.how to apply them correctly”.

The unit explores the rules for manipulation of formulae and equationsThe unit explores the rules for manipulation of formulae and equations

Aims….Aims….

Aims for this topic:Aims for this topic:

You will be Learn how to solve real problems by forming You will be Learn how to solve real problems by forming and solving simultaneous equations using an algebraic and solving simultaneous equations using an algebraic methodmethod

You will learn how to solve simultaneous equations using You will learn how to solve simultaneous equations using graphsgraphs

On completion of this unit a learner should:On completion of this unit a learner should:

Know the basic underpinning mathematical techniques- Know the basic underpinning mathematical techniques- methods used to manipulate and/or solve formulae, methods used to manipulate and/or solve formulae, equations and algebraic expressionsequations and algebraic expressions

Recap….

SimultaneousEquations recap

How do you go about solving simultaneous equations?...

A few hints . . .

(1)Scale up each term in one, or both equations to make the same number in front of either the x terms or the y terms.

(2)Subtract if the signs in front if the signs are the same.

(3) Add if the signs in front of the numbers are different.

Lets work through the first one together – have your NO MESS handout ready…..

5x + y = 20

2x + y = 11

… (1)

… (2)

Numberthe Equations

3x = 9Subtract

(to get rid of a letter)

Divide(to find x)

x = 3

Substitutein (2)

(to find y)

2 x 3 + y = 11

6 + y = 11

y = 5

1

You try the next one…

7x + y = 43

3x + y = 23

… (1)

… (2)

Numberthe

Equations

4x = 20Subtract

(to get rid of a

letter)

Divide(to find x)

x = 5

Substitutein (2)

(to find y)

3 x 5 + y = 23

15 + y = 23

y = 8

2

A nice System….

when using the elimination method to solve when using the elimination method to solve

equations equations FOR NO MESS FOR NO MESS RememberRemember

FORFORM your equationsM your equations

NNeatlyeatly

OOrganise your equations rganise your equations aax x + b+ byy = number = number

MMake the number of x’s or y’s the same if you have toake the number of x’s or y’s the same if you have to

EEliminate a letter liminate a letter

SSolve the equationolve the equation

SSubstituteubstitute

CCheckheck your answer! your answer!Next stage - Form your own…

Forming equationsForming equations

Next we will look at forming our own Next we will look at forming our own equations from given information-turning equations from given information-turning word problems into equationsword problems into equations

We will do the first task We will do the first task

together…together…

Task 1: Financial Penalty clauseTask 1: Financial Penalty clause A penalty clause states that a contractor will A penalty clause states that a contractor will forfeit a certain sum of money for each day that forfeit a certain sum of money for each day that they are late in completing a contractthey are late in completing a contract

(i.e. the contractor gets paid the value of the (i.e. the contractor gets paid the value of the original contract less any forfeit). original contract less any forfeit).

If they are 6 days late they receive £5000 If they are 6 days late they receive £5000

if they are 14 days late they receive £3000.if they are 14 days late they receive £3000.

(a) Find the amount of the daily forfeit(a) Find the amount of the daily forfeit (b) Determine the value of the original contract.(b) Determine the value of the original contract.

How?…

Task 1Task 1 Solution SolutionTo solve this problem - there are 2 unknowns…..To solve this problem - there are 2 unknowns…..the original contract amount Cthe original contract amount Cthe Daily forfeit D the Daily forfeit D

For 6 Days late C+6D = 5000 For 6 Days late C+6D = 5000 (1)(1)For 14 Days late C+14D = 3000 For 14 Days late C+14D = 3000 (2) Then use (2) - (1) (2) Then use (2) - (1) 8D = -20008D = -2000 D = -D = -250 250

Sub D Sub D intointo (1) (1) gives C + 6 (-250) = 5000gives C + 6 (-250) = 5000

C-1500 = 5000C-1500 = 5000

C = C = £6500£6500 You need a Strategy – see NO MESS handout….

StrategyRemember to use the phrase… Remember to use the phrase…

FOR NO MESSFOR NO MESS

-Now lets try one…

Task 2: General Office equipmentTask 2: General Office equipment

The total cost of equipping 2 offices, A The total cost of equipping 2 offices, A and B is £30,000. and B is £30,000.

Office B costs £2,000 more than office A. Office B costs £2,000 more than office A.

find the cost of equipment for each of find the cost of equipment for each of them.them.

Task 2: Task 2: SolutionSolution

A+B = 30,000 A+B = 30,000 (1)(1)A-2000 = B A-2000 = B (2)(2)Re-arrangeRe-arrange (2) giving (2) givingA-B = 2000 A-B = 2000 (3) (3)

(1) - (3) (1) - (3) leaves B- -B = 28000leaves B- -B = 28000 2B = 280002B = 28000 B = £14000B = £14000 Then A = £16000Then A = £16000

Task 3: Office heating: costsTask 3: Office heating: costs

A heating installation for one office consists of 5 radiators A heating installation for one office consists of 5 radiators and 4 convector heaters and the cost including labour is and 4 convector heaters and the cost including labour is £2080.£2080.

In a second type of office 6 radiators and 7 convector In a second type of office 6 radiators and 7 convector heaters are used and the cost, including labour is £3076.heaters are used and the cost, including labour is £3076.

In each office, installation costs are £400.In each office, installation costs are £400.

Find the cost of a radiator and the cost of a convector Find the cost of a radiator and the cost of a convector heater.heater.

Task 3: Task 3: SolutionSolution

5r + 4c = 2080 - 400 5r + 4c = 2080 - 400 (1)(1)6r + 7c = 3076 - 400 6r + 7c = 3076 - 400 (2)(2)5r +4c = 16805r +4c = 1680 (1) (1) 6r + 7c = 26766r + 7c = 2676 (2) (2)(1) x 6/5 (1) x 6/5 gives 6r + 4.8c = 2016gives 6r + 4.8c = 2016 (3) (3) (2) –(3) (2) –(3) 2.2c = 6602.2c = 660 C = £300C = £300Sub cSub c into into (1)(1)5r + 4 x 300 = 16805r + 4 x 300 = 1680r = £480r = £480

Task 4: Plumbing costsTask 4: Plumbing costs

100m of tubing and 8 elbow fittings 100m of tubing and 8 elbow fittings cost £100.00.cost £100.00.

150m of tubing and 10 elbow fittings 150m of tubing and 10 elbow fittings cost £147.50.cost £147.50.

Find the cost of 1m of tubing and the Find the cost of 1m of tubing and the cost of an elbow fitting.cost of an elbow fitting.

Task 4: Task 4: SolutionSolution 100m + 8e = 100 100m + 8e = 100 (1)(1) 150m + 10e = 147.50 150m + 10e = 147.50 (2)(2)(1) x 1.5 gives 150m + 12e = 150 (1) x 1.5 gives 150m + 12e = 150 (3)(3)

(3) – (2) gives 2e = 2.50(3) – (2) gives 2e = 2.50 e = £1.25e = £1.25

Sub e into (1) 100m + 8x£1.25 = 100Sub e into (1) 100m + 8x£1.25 = 100 100m + 10 = 100100m + 10 = 100 100m = 90100m = 90 90/100 = m 90/100 = m m= 90pm= 90p

Task 5: Orders-Pricing-inflation Task 5: Orders-Pricing-inflation

The cost of grade B bolts is £0.60pThe cost of grade B bolts is £0.60p

The cost of grade A bolts is £0.90p. The cost of grade A bolts is £0.90p.

A combination of several samples of each costs A combination of several samples of each costs £9.00. £9.00.

After a year into the construction, inflation raises the After a year into the construction, inflation raises the cost of each sample by 15p and the cost of the cost of each sample by 15p and the cost of the combination rises to £ 10.65.combination rises to £ 10.65.

Find the number of each sample ordered.Find the number of each sample ordered.

Task 5: Task 5: SolutionSolution

0.60B + 0.90A = 9.00 (1)0.60B + 0.90A = 9.00 (1) 0.75B + 1.05A = 10.65 (2)0.75B + 1.05A = 10.65 (2)(1) x 0.15 gives 0.75B + 1.125A = 11.25 (3)(1) x 0.15 gives 0.75B + 1.125A = 11.25 (3) (3) – (2) gives 0.075A = 0.60 (3) – (2) gives 0.075A = 0.60 0.60/0.075 = 0.60/0.075 = 88 = A = ASub A into (1) gives 0.60B + 0.90 x 8 = 9Sub A into (1) gives 0.60B + 0.90 x 8 = 9 0.60B + 7.2 = 90.60B + 7.2 = 9 0.60B = 1.80.60B = 1.8 1.8/0.60 = 1.8/0.60 = 3 = B3 = B

Task 6: Construction plantTask 6: Construction plant

For the crane which lifts the items it is found that the For the crane which lifts the items it is found that the effort E Newtons and the load W Newtons are effort E Newtons and the load W Newtons are connected by the equation E =aW + b. connected by the equation E =aW + b.

An effort of 90N lifts a load of 100NAn effort of 90N lifts a load of 100N

An effort of 130N lifts a load of 200 N.An effort of 130N lifts a load of 200 N.

Find the values of a and b Find the values of a and b

Determine the effort required to lift a load of 300NDetermine the effort required to lift a load of 300N

Task 6: Task 6: SolutionSolution E = aW + bE = aW + b

90 = 100a + b (1)90 = 100a + b (1)

130 = 200a + b (2)130 = 200a + b (2)

(2) - (1) gives: 40 = 100a (2) - (1) gives: 40 = 100a

a=0.4a=0.4

Sub a into (1) gives:90 = 100x0.4 + bSub a into (1) gives:90 = 100x0.4 + b

90 = 40 + b90 = 40 + b

b = 50b = 50

Now use E = aW + bNow use E = aW + b

E = 0.4 x 300 + 50E = 0.4 x 300 + 50

E = 170NE = 170N