Lesson video

Hi, I'm Mrs. Wheelhouse, and welcome to today's lesson that's from the unit, Maths in the Workplace.

In this series of lessons, we're exploring how maths is used in different careers.

So let's get started.

In today's lesson, we're going to be looking at some of the ways maths is used in the construction industry.

Let's get started.

On the screen, there are some keywords we're gonna be using in our lesson today.

Now, these should be familiar to you, but if they're not, feel free to pause the video and take a moment to read through them all.

Our lesson today is broken into two parts, and we're going to begin by looking at how formulae is used in the construction industry.

What areas of maths do you think a construction worker might use? Pause the video while you have a discussion now.

Well, this is just some of the ways I came up with.

So I thought maybe area and volume, plans and elevations, ratio, percentages, money, measures, proportion, formulae, and standard form.

Now in this lesson we're going to look at some of these areas.

We couldn't explore them all, we just don't have time.

So you might find it hard to believe that many formulae are used in the construction industry.

Can you think of any formulae that might be used? Well, here are some examples.

We have total concrete quantity is the volume of the concrete multiplied by 1.

54.

We have a slope as a percentage, which is the vertical distance divided by the horizontal distance and multiplied by 100.

Hmm, vertical distance divided by horizontal distance.

I think you might have met that before.

Can you think where? That's right when you were calculating gradients.

Number of bricks is equal to the wall volume multiplied by 1,000, and that's divided by the brick volume and the mortar volume sum together.

Ooh, height is equal to 2.

304 multiplied by pressure.

I wonder what that's looking into.

There's also Ohm's Law.

We'll explore that one a little bit, as well as Joule's Law.

Now let's look at just the first three to start.

Which type of construction worker do you think uses these formulae? It's a builder.

Which type of construction worker do you think might use this formula? It's a plumber.

And what about the last two? That's right, it's an electrician.

So here's another example of a formula used by builders.

In this we're looking at length of rafter squared is equal to the rise of the rafter squared plus the run of the rafter squared.

Does that look familiar to you? Pause the video and have a quick discussion now.

Welcome back.

That's right, it's Pythagoras' theorem.

Do you remember that? C squared is equals A squared plus B squared.

Ah, it might look familiar now.

There we go.

Here are the front and side elevations of a shed.

The builder needs to replace the rafters of the shed.

How long are the rafters? Well, that's where a rafter would be.

And of course we can draw on a right-angled triangle.

To calculate the length of A, we must take the length of the entire shed and divide it by two.

And that gives us 1.

19.

For B, I need to look at the other drawing of the shed and I can see the entire height of the building is 2.

31.

Remember, I only want the height from where the rafter begins to where the rafter ends.

So I'm gonna subtract the 1.

8 and that gives me the difference.

So B is 0.

51.

I can now calculate the length of the rafter.

C, the length of the rafter, is equal to 1.

295, and that's to three decimal places, and that length is in metres.

Your turn now.

Here are the front and side elevations of the shed.

The builder needs to replace the rafters of the shed.

How long are the rafters? Pause the video and work this out now.

Welcome back.

You should have got that A is equal to 3.

2 divided by two, in other words, 1.

6, and that B is equal to 2.

4 subtract two, so it's 0.

4.

Then putting these into Pythagoras' theorem should have got that C is equal to, now I've written 1.

65 metres 'cause I've written it to two decimal places.

As long as your answer rounds to this, that's absolutely fine.

Now let's have a look at a formula that plumbers use.

Now they a formula to calculate flow rate.

They use the formula Q equals A multiplied by V.

Where Q is the flow rate, A is the flow area, and V is the velocity.

Let's look at how we could use that formula.

Water runs through a cylindrical pipe at a velocity of 10 metres per second.

The diameter of the pipe is 25 millimetres.

What is the flow rate in cubic metres per hour? Well, what shape is the flow area? That's right, it's a circle.

So let's calculate the area of the cross-section of the pipe to two significant figures.

Having done that, I can now calculate the flow rate.

I take the flow area and multiply it by the velocity.

Now, because I wanted it per hour, you'll notice I didn't just multiply it by 10 because this was 10 metres per second.

So I needed to convert from seconds to hours, which is why I'm multiplying by 3,600.

When I put this into my calculator, I got a flow rate of 17.

67 cubed metres per hour.

Quick check you understand that.

Plumbers use the formula below to calculate flow rate.

That's the same formula you saw before.

Water's running through a cylindrical pipe of diameter 32 millimetres at a velocity of 10 metres per second.

Which of the following is the correct calculation for the flow rate in cubed metres per hour? Is it A, B, or C? Pause the video while you make your choice now.

Welcome back.

You should have gone with B.

Well done if you did.

Electricians use Ohm's Law to determine the voltage resistance or current of an electric circuit.

So we have V equals I multiplied by R, where V as voltage, I is current, and R is resistance.

They also use a formula to calculate power.

P equals I times V, where P is the power, I is the current, so exactly the same as it was for the previous formula, and V is voltage, just like it was in the previous formula.

So let's see how we could use these.

A 120 volt 20 amp circuit is used to supply lampposts for a carpark.

How many lampposts can be safely installed if the bulbs are 400 watts? Well, let's use the formula, P equals I times V.

Remember, I is the current, so we have 20 multiplied by the voltage, 120, gives us a power of 2,400 watts.

So the number of lampposts we can safely instal is 2,400 divided by 400, or six lampposts.

Now generally, power is given in kilowatts.

The power of a kettle is two kilowatts.

How many watts is that? Did you spot there might be a hint? Look at the kilowatt, what does that tell you? Well, the prefix kilo represents 1,000, remember? So what is that power of a kettle in watts? It's 2,000 watts.

Your turn now.

Please calculate the power of the following household appliances in watts.

Pause the video while you do this.

Welcome back.

We should have multiplied them, in most cases, by 1,000, but did you spot there was a little trick in there? So the shower is 8,000 watts, the water tank heater is 3,000 watts, the light bulb is 60 watts.

We didn't have to change it 'cause the unit was already correct.

The vacuum cleaner is 900 watts, the fan heater, 2,400 watts, the hairdryer, 2,200 watts, and the microwave, 1,500 watts.

Well done.

Now, the voltage of electricity in UK households is 230 volts.

An electrician needs to instal an eight kilowatt shower.

Circuits are available in six, 10, 16, 20, 32, 40, 50, and 63 amps.

Which circuit should be installed? Well, to work this out, we're going to need some formulae.

We need P equals I times V.

We've got the 8,000 watts, which is our shower, is equal to I multiplied by 230, 230, remember, is the voltage of electricity in UK households.

So by calculating the division, we've got that the current needs to be 34.

8 amps, and that's to one decimal place.

So I need to make sure that the circuit can handle that, which means I need to instal a 40 amp circuit.

An electrician is testing a two kilowatt kettle for a fault.

The manufacturer's information shows the resistance of the kettle as 25 ohms. Is the kettle faulty? Well, let's use a formula to check.

So I started with P equals I times V.

The power, remember, is 2,000, and the voltage of the household is 230.

So that's allowed me to work out the current.

And this will let me work out the resistance.

The resistance is 26.

45 ohms. This is very close to the expected value so it's unlikely that our kettle is faulty.

Now, the voltage of electricity in UK households is 230 volts.

Therefore a six amp circuit should be installed for a 1.

5 kilowatt microwave.

Now, I've put a formula up there to help you.

Is this true or is this false? Pause the video now while you work it out.

It's false.

Do you have your working to justify why? That's right.

I can rearrange that formula to give I is equal to P divided by V.

And I, therefore, is 6.

52 amps to two decimal places.

The current is higher than six amps, therefore a six amp circuit is not going to be sufficient.

Now, plumbers use formula to calculate the most appropriate size of radiator required for a room.

This is based on the size of the room, the temperature the room needs to be heated to, and the areas where heat is lost, for example, windows.

Below is the formula to calculate BTU, which is British Thermal Units.

So in order to calculate the BTU, we need the volume of the room and we multiply it by the allowance for heat loss.

And what I've done is produce a table for you where you can see different rooms and the allowance for heat loss.

So below are the dimensions of the living room, we're going to calculate the BTU.

Well, we need the volume of the room, so let's calculate that.

So that's 7.

6 times 6.

3 times 2.

3.

And then we multiply it by the allowance for heat loss.

Well, it's a living room so I can multiply by 135.

This gives me that the BTU is 15,000, to two significant figures.

So that's what I need.

So what's the most appropriate radiators for this room? Well, you can see different radiator dimensions and their BTUs.

Well, let's assume this room will have two radiators, which makes sense because the largest radiator does not have a BTU, which is an excess of 15,000.

So that means I could look for two radiators of around 7,500 each.

Well, that means I could use two of the 600 by 1,400 radiators.

They'd be quite appropriate for the size of the living room.

And that way I've used radiators of the same size, which would be aesthetically pleasing.

Your turn now.

The required BTU for a kitchen is 3,208.

Which size of radiator should be used? So assume I'm using just one radiator here.

Pause the video and work this out now.

Welcome back.

You should have chosen the 600 by 600 radiator.

It's time for your first task.

For question one, I'd like you to work out how long the rafters are that the builder needs.

Pause and do this now.

For question two, I'd like you to calculate the flow rate in cubed metres per hour please.

Pause and do this now.

Question three, I'd like you to calculate which circuit should be installed.

Pause and do this now.

For question four, using the information on the screen, I'd like you to calculate the BTU.

And then for part B, the bedroom's going to have one radiator, tell me which size should be installed.

Pause and do this now.

Welcome back.

Time to go through our answers.

So for question one, the rafters to three decimal places, have a length of 2.

062 metres.

So in other words, I've calculated this to the nearest millimetre, but as long as your answer either rounds to this or is close to this, that's acceptable.

So the flow rate should be 45.

24 cubed metres per hour, and that's to two decimal places.

For question three, you should have calculated that the current was 9.

35 amps, which means a 10 amp circuit should be installed.

For question four, the BTU was 3,100.

032.

So in other words, if I'm going to have one radiator, what size should be installed? That's the 600 by 600.

Well done if you've got these all right.

It's now time for the second part of our lesson, and that's costing in the construction industry.

So let's go back to our previous problem where the builder was replacing the rafters of the shed.

They need to replace eight rafters.

Well, we worked out that each rafter was 1.

295 metres in length.

So how much will it cost to purchase the eight lengths I need? Well, can't I just find the total length of the eight rafters and then use that length? What do you think? No, I can't because the full lengths of each rafter need to be in one piece.

I don't want one of my rafters to be broken into lots of small pieces, that's no use to me.

So let's calculate the number of rafters we can make from each length of wood.

Well, first I need to make sure all of our measurements are in the same units.

So let's convert them all to millimetres.

There we go.

Now each rafter is 1,295 millimetres in length, and that's why I've given my answer to three decimal places before, just in case I needed to convert to millimetres, and I didn't lose any accuracy.

So if I consider the length of wood that's 2,400 millimetres, mm, I can only get one rafter out of that length of wood.

For 3,000 millimetres, I can get two rafters.

3,600 produces two rafters.

The 4,200 though produces three.

And the 4,800 produces three.

If I'm buying two rafters, which would be better value for money? The 3,000 millimetre length or the 3,600 millimetre length? Well, the table makes that really clear, it's the 3,000 millimetre length.

So in other words, there's no option where I'm gonna choose to buy the 3,600 millimetre length, it's just not as good value for money.

So let's get rid of that.

If I'm buying three rafters, what's better value for money? The 4,200 millimetre length or the 4,800 millimetre length? Exactly, it's the 4,200.

So it gets rid of another option.

So let's consider all the different options I have for buying eight rafters.

Well, I could buy three lots of the 4,200 millimetre length, which means that I'd have one spare rafter, but I've definitely got my eight and some spare wood.

I could buy two lots of the three and one lot of the two.

That makes eight exactly.

Now there are other options, but I know that one of these is gonna be the most cost effective.

Can you spot why? Well, you should have spotted that buying one rafter twice is easily more expensive than buying the length of wood that will give me two rafters.

And buying one rafter three times so that I've got three rafters is definitely more expensive than just buying the length of wood that will generate three rafters already.

So here are my two options along with how much each one costs.

We can see that that's the best option therefore, buying two lengths of 4,200 millimetres and one length of 3,000 millimetres for a total cost of £19.

95.

Right, quick check.

The cheapest way of making eight rafters each of length 1.

5 metres is £21.

Now is that true or is that false? Pause the video while you work this out.

Welcome back.

You should have said it's true.

Two rafters can be cut from the 3,000 millimetre length with no wastage whatsoever.

So if I buy four of those, four lots of £5.

25 is £21.

It's now time for your final task.

For question one, the table's showing the prices and lengths the wood is sold in.

Calculate the cheapest price for purchasing the following: Part A, 20 one-meter lengths.

Part B, 15 1.

8-meter lengths.

Part C, three different ways to buy 600 1.

2-meter lengths.

And D, why are there three different ways for Part C? Pause the video while you work this out now.

Let's go through those answers.

So to buy the 20 one-meter lengths, well, the 3,000 millimetre length produces three one-meter lengths.

So six lots of those plus one lot of the 2,400 millimetre length will give us 20 one-meter lengths, a total cost of £35.

70.

Part B, the 3,600 millimetre length produces two 1.

8 metre lengths.

So seven of these plus one of the 2,400 millimetre length will give us 15 1.

8-meter lengths at a total cost of £48.

30.

Part C, here are three different ways to buy 600 1.

2 metre lengths.

I could buy 300 of the 2,400 millimetre lengths, I could buy 200 of the 3,600 millimetre lengths, or I could buy 150 of the 4,800 millimetre lengths.

Notice that all of these cost us £1,260.

So why are there three different ways? Well, that's because the 2,400, 3,600, and 4,800 millimetre lengths each produce no wastage and the cost per metre of wood is the same for all three.

Well done if you spotted that.

Let's sum up what we've looked at today.

Many areas of the construction industry use formulae.

Units and accurate measurements are essential to ensure things like safety.

Costing is another really important factor of the construction industry.

Wastage is also considered when purchasing materials for jobs.

Well done, you've worked really well today.

And I hope you've enjoyed learning some of the ways that maths is used in the construction industry.

I look forward to seeing you again for some more maths.

Bye, for now.