The situation
Headquarters has been told that a huge storm has damaged the research base in the Arctic. Luckily, the rescue team from our bases in the area managed to rescue the scientists who were working there, and they are expected to recover. You are the new team who must rebuild the laboratory, carry on with their research and keep yourselves safe.
You have limited resources, extreme conditions and very little time to make critical decisions. You will have to calculate, plan, optimise and justify every step with scientific and mathematical rigour.
Note. All calculations are rounded to two decimal places.
Prior knowledge
- Geometry and measurement: areas of plane figures (square, rectangle, triangle, circle); base area of solids; volumes: prism $V = A_{\text{base}} \cdot h$, cylinder $V = \pi r^2 h$, pyramid $V = \frac{1}{3} A_{\text{base}} \cdot h$.
- Units: length, area and volume units, and how they relate to litres: $1\ \text{dm}^3 = 1$ L and $1\ \text{m}^3 = 1000$ L.
- Arithmetic and proportion: rule of three, percentages and operations with decimals and with $\pi$.
- Equations: turning a statement into an equation, solving for the unknown (for example $a^2 = 12$) and square roots.
- Context: interpreting the results (litres, days, free space) and checking that they make sense.
Challenge 1: Designing the research camp
We realise that every module is unusable, although most of the furniture and materials inside can be reused. We have to redesign the camp and, given the urgency, these are the only modules we have:
Square-based prism
2,5 × 2,5 × 6 m
Buildings
Large cylinder
radius 1.2 m · height 2.5 m
Drinking water
Square-based pyramid
3,5 × 3,5 × 2,5 m
Fuel · recyclable waste
Small cylinder
radius 1 m · height 2 m
Organic waste · general waste
For now there will be 12 people at the camp, and the total area available is 300 m². We need:
Modules for the camp buildings
- 1 sleeping module for every 3 people.
- 1 kitchen module.
- 1 laboratory module for every 4 people.
- 1 storage module for every 6 people.
Extra modules in total
- 1 drinking water module.
- 1 fuel module.
- 1 organic waste module.
- 1 general waste module.
- 1 recyclable waste module.
Task 1. Do all the modules fit?
Before going on, check whether all these modules can be placed in the camp; if not, some things will have to go. Step 1: work out how many modules of each type you need. Step 2: work out the base area of each one. Step 3: find the total area and whether they all fit.
See the solution
Building modules (prisms), for 12 people:
Extras: 1 large cylinder (water), 2 pyramids (fuel and recyclables) and 2 small cylinders (organic and general waste).
10 prisms + 1 large cylinder + 2 pyramids + 2 small cylinders = 15 modules
The base of the prisms and pyramids is a square ($\ell^2$) and that of the cylinders is a circle ($\pi r^2$):
| Number | Shape | Base area (m²) | Subtotal (m²) |
|---|---|---|---|
| 10 | Square-based prism | $2{,}5^2 = 6{,}25$ | $62{,}50$ |
| 1 | Large cylinder | $\pi \cdot 1{,}2^2 \approx 4{,}52$ | $4{,}52$ |
| 2 | Square-based pyramid | $3{,}5^2 = 12{,}25$ | $24{,}50$ |
| 2 | Small cylinder | $\pi \cdot 1^2 \approx 3{,}14$ | $6{,}28$ |
That is far less than the $300\ \text{m}^2$ available.
If the cylinder bases are only rounded at the end, the total is $97{,}81\ \text{m}^2$.
The modules take up $97{,}80\ \text{m}^2$: they all fit.
Task 2. Free space
Work out what percentage of the camp is left free of modules. For safety it must be more than 30 %. Is it?
See the solution
With $97{,}81\ \text{m}^2$ there are $202{,}19\ \text{m}^2$ free, and the percentage is also $67{,}40\ \%$.
$67{,}40\ \%$ is left free, more than $30\ \%$: the camp is safe.
Challenge 2: Checking capacities
In the middle of the Arctic we must be sure we have enough oxygen if we have to seal ourselves in, enough drinking water, and enough room for waste, which can only be emptied when it is collected. To know the capacity we need the volume of each module, remembering that $1\ \text{dm}^3 = 1$ L.
Task 1. The capacity of the modules
Work out the capacity of the modules, in m³ and in litres.
See the solution
Since $1\ \text{m}^3 = 1000\ \text{dm}^3 = 1000$ L, we multiply by $1000$ to get litres. Volume of each module:
| Number | Shape | Calculation | Volume (m³) | Litres |
|---|---|---|---|---|
| 10 | Square-based prism | $2{,}5 \cdot 2{,}5 \cdot 6$ | $37{,}50$ | $37\,500$ |
| 1 | Large cylinder | $\pi \cdot 1{,}2^2 \cdot 2{,}5$ | $11{,}31$ | $11\,310$ |
| 2 | Square-based pyramid | $\tfrac{1}{3} \cdot 3{,}5^2 \cdot 2{,}5$ | $10{,}21$ | $10\,210$ |
| 2 | Small cylinder | $\pi \cdot 1^2 \cdot 2$ | $6{,}28$ | $6\,280$ |
A pyramid has a third of the volume of the prism with the same base and height; that is where the $\tfrac{1}{3}$ comes from.
Task 2. The air in the buildings
How much air is there in the building modules?
See the solution
The buildings are the 10 prisms:
$375\,000$ L of air
Task 3. Water, fuel and waste
How much drinking water, fuel and waste of each type can we store?
See the solution
| What | Module | Capacity |
|---|---|---|
| Drinking water | 1 large cylinder | $11\,310$ L |
| Fuel | 1 pyramid | $10\,210$ L |
| Recyclable waste | 1 pyramid | $10\,210$ L |
| Organic waste | 1 small cylinder | $6\,280$ L |
| General waste | 1 small cylinder | $6\,280$ L |
Challenge 3: Survival
Before sending the expedition we must check that we meet the survival requirements.
A. Oxygen during a lockdown
In an extreme storm or a chemical leak, the 12 people will have to shut themselves in the building modules, which are connected to each other. We want to know how long they could last on the oxygen without renewing the air.
Data: a person at rest needs about $0{,}5$ L of oxygen per minute, and the air we breathe is $21\ \%$ oxygen.
Task A1
Using the data from challenge 2, how much air is there in the 10 modules?
See the solution
$375\,000$ L
Task A2
How much oxygen is there in that air?
See the solution
$21\ \%$ of the air is oxygen:
$78\,750$ L of oxygen
Task A3
How much oxygen do the 12 people need every minute?
See the solution
$6$ L/min
Task A4
Work out how many minutes we have before the oxygen runs out, and convert it to hours and days.
See the solution
$13\,125$ min $= 218{,}75$ h $\approx 9{,}11$ days
Task A5
How many days will they last? Do you think survival depends only on oxygen?
See the solution
Going by oxygen, about 9 days. But it does not depend only on oxygen:
- The carbon dioxide we breathe out builds up and becomes dangerous long before the oxygen runs out.
- Furniture and materials take up part of the modules: there is less air than the volume we calculated.
- Water, food, heating with the fuel and humidity also matter.
About 9 days, but the real limit would come sooner.
B. A smaller fuel pyramid
We would like to make the fuel pyramid smaller, because we will not need that much and it is an extra hazard. Can you find an equation that gives a new side length for the square base so that the fuel is exactly $10\ \text{m}^3$? The height is still $2{,}5$ m.
See the solution
Volume of a square-based pyramid with side $a$ and height $h$:
With $V = 10$ and $h = 2{,}5$, the equation is:
We solve for $a^2$ and take the square root (the side must be positive):
Check: with $a = 3{,}46$ we get $V \approx 9{,}98\ \text{m}^3$; the difference comes from rounding. Notice the side only shrinks by 4 cm, from $3{,}5$ to $3{,}46$ m: the original volume, $10{,}21\ \text{m}^3$, was already very close to 10.
The side of the base must be $\sqrt{12} \approx 3{,}46$ m.