The Modelling Cycle (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Christopher is estimating how many trees are in a wood near where he lives.

Describe how he might work out an estimate for the number of trees in the wood.

Essential factors:
Size (area) of the wood
How densely the trees are packed in

Simplified factors:
Assume the density of trees is the same throughout the wood
Assume a rough shape for the wood, e.g. a circular or a rectangular area

Ignore:
Different densities of trees in different areas (due to different species etc.)
Technicalities about what counts as a "tree" and what does not

Rounding:
Depending on size of wood, nearest hundred or nearest thousand is probably appropriate

Christopher could use an online mapping tool to view the general shape of the wood, and approximate it to a simple shape such as a circle, rectangle, or triangle. He could then use the scale shown on the online mapping tool to calculate the area of the wood.

Go to a section of the wood in-person and take a sample of the density of trees. For example, count how many trees are there in a 10 m × 10 m square. Use this density, and the calculated area, to estimate the number of trees in the wood.

Omar has a model truck. It is a 1:76 scale replica of a real truck.
Omar finds that he can fit around 100 of his marbles, which are spheres with a diameter of 1 to 2 cm, into the trailer of the model truck.

Find an estimate for the volume of the trailer for the real truck, to the nearest ten thousand litres.

The key pieces of mathematical understanding required here are:

• Finding the volume of a sphere;

• Relating a linear scale factor to a volume scale factor; the volume scale factor is the cube of the linear scale factor

• Converting between cm³ and litres; there are 1000 cm³ in 1 litre

Find the volume of a marble, assume the diameter is in the middle of the range, so the radius will be around 0.75 cm

Volume of a marble =

Round to 1.77 cm³ for ease of calculation; we only need an estimate

Find total volume of the model truck's trailer

1.77 cm³ × 100 marbles = 177 cm³

The linear scale factor is 76, as the model uses a 1:76 scale
Therefore the volume scale factor will be 76³

76³ × 177 cm³ = 77 698 752 cm³

Convert to litres by dividing by 1000

77 698 752 cm³ ÷ 1000 = 77 698.752 litres

Round to the nearest ten thousand litres

80 000 litres

Nell is painting several rooms of her house. The total surface area she is planning to paint has been previously calculated to be 148 m².

The paint that Nell is using requires 2 coats.
It is sold in tins of either 2.5 litres for £22, or 5 litres for £34.
There is also a "buy 4, get the 5th half price" special offer on the 5 litre tins of paint.
The paint states it covers up to 14 m² per litre.

Calculate the minimum cost for the tins of paint that Nell needs to buy.

Calculate the total surface area to be covered

148 m² × 2 coats = 296 m²

Divide this by the coverage to find the volume of paint required

296 m² ÷ 14 m² per litre = 21.14... litres

The paint is only sold in 2.5 litre or 5 litre tins, so round upwards to nearest 0.5 litres
This step is an example of interpreting a result in context

21.5 litres of paint required

Consider if it is cheaper to purchase 2.5 litre or 5 litre tins

2.5 litres for £22 = £8.80 per litre
£34 for 5 litres = £6.80 per litre

So minimise the number of 2.5 litre tins bought, as they are more expensive

21.5 litres = (4×5 litres) + 1.5 litres

So purchase 4 × 5 litre tins, and 1 × 2.5 litre tin

Total cost = (4 × £34) + £22 = £158

However, using the extra context given in the problem about the special offer:
"buy 4, get the 5th half price" on all 5 litre tins

So it would be cheaper to purchase a half-price 5 litre tin, than a single 2.5 litre tin

Total cost = (4 × £34) + (0.5 × £34) = £153

£153

Ruben is an ice-cream seller. He estimates that he sells around 50 ice creams per day over the course of one 100-day season. The ice creams are each estimated to be 100 ml per serving.

Ruben is visiting his supplier before the season begins to purchase the ice-cream he needs.

He wants to purchase as much as possible in one transaction as the supplier offers a discount proportional to the volume of ice-creams purchased. It is sold in 12 litre containers.

(a) Advise Ruben on how many containers of ice-cream he should purchase from his supplier.

Find the total volume sold over the season

50 ice creams per day × 100 days × 100 ml = 500 000 ml

Convert to litres

500 000 ÷ 1000 = 500 litres

Find the number of containers

500 ÷ 12 = 41.666...

Round up, as requires an integer number of containers

Ruben should buy 42 containers of ice-cream for the season

(b) Suggest a reason why buying all of the ice cream at once might not be a good idea.

It is a very large volume of ice cream, and Ruben would need somewhere to store it all which may not be practical

As it is ice-cream, it will require freezing too, which will cost money as energy is required

Monique is calculating how many moving boxes will fit in the back of her removals van.

The storage area of the van is 4.2 m long, 2.1 m wide, and 2.2 m high.

The moving boxes each measure 50 cm in length and are cube-shaped.

Monique does the following calculations:

Volume of moving box: 0.5 × 0.5 × 0.5 = 0.125 m³

Volume of van: 4.2 × 2.1 × 2.2 = 19.404 m³

Number of boxes per van: 19.404 ÷ 0.125 = 155.232

Therefore 156 boxes will fit in the van

(a) Evaluate Monique's method and answer.

Work through each step of the method and check the techniques chosen are appropriate, and the calculations have been performed correctly

The volume of the moving boxes and the van have both been calculated correctly

Consider any real-life restrictions which have, or have not been, included

When calculating the number of boxes per van, the answer has been rounded up rather than down

As only an integer number of boxes is valid, the answer using this method should be 155 instead

Consider any restrictions which have not been taken into account, or incorrect assumptions
State the effect these have on the answer

The real dimensions of the boxes have not been taken into account

As the boxes must remain in the same dimensions, and half-boxes or similar cannot be used to fill in gaps, this limits the number of boxes that will fit in the van

This has led to an overestimate

(b) Make one improvement to the model and find a new estimate.

The moving boxes measure 0.5 m in length, width and height, and we can take this into account when finding how many will physically fit in the van

Considering the length of the van
4.2 ÷ 0.5 = 8.4 boxes, so 8 boxes maximum

Considering the width of the van
2.1 ÷ 0.5 = 4.2 boxes, so 4 boxes maximum

Considering the height of the van
2.2 ÷ 0.5 = 4.4 boxes, so 4 boxes maximum

Find the total number of boxes

8 × 4 × 4 = 128

128 boxes

(c) Moving boxes are sometimes too heavy to be stacked on top of one another. What effect does this have on the number of boxes able to fit inside the van?

The number of boxes would reduce, as the previous solution relied on 4 boxes being stacked on top of one another