Excellence: assumptions, limitations and general rules
What this lesson covers
- Every model rests on assumptions, and changing one can change the answer.
- Extending a problem means asking what happens in general, or what the model cannot tell you.
Excellence means extending the problem
- Take at least one problem further than asked.
- Four ways to do it.
- All four, on one rainwater tank.
| Extend by | Means |
|---|---|
| Assumptions | what the model takes for granted |
| Limitations | what it cannot tell you |
| Generalising | a rule with a variable |
| Predicting | using the rule for a new case |
The model so far
- A roof collecting into a cylinder tank.
- 147 mm of rain fills it.
- A full tank lasts 78 days.
| Result | |
|---|---|
| Roof | 96 m² collecting area |
| Tank | 14 137 litres |
| Rain to fill | 147 mm |
| Lasts | 78 full days at 180 L a day |
What the model assumes
- Every answer rests on assumptions.
- Write them down, one by one.
- Say which ones are doubtful.
| Assumption | But really |
|---|---|
| Every drop is collected | gutters and splash lose some |
| Use is always 180 L | washing days, visitors |
| No rain while it empties | rarely true |
| No overflow | a full tank spills |
Test one: only 80% collected
- Suppose gutters and splash lose a fifth.
- The roof now collects 76.8d cubic metres.
- 184 mm of rain, not 147.
| Working | ||
|---|---|---|
| Collected | 0.8 × 96d = 76.8d | |
| Equation | 76.8d = 14.137 | |
| Rain needed | d = 0.184 m | 184 mm |
| Was | 147 mm | 37 mm more |
How sensitive is the answer?
- Collecting 80% means needing 1.25 times the rain.
- A 20% loss costs 25% more rain.
- That is worth saying in the conclusion.
| Water collected | falls to 80% |
| Rain needed | 1 ÷ 0.8 = 1.25 times as much |
| Check | 147 × 1.25 ≈ 184 mm |
A general rule
- Replace the fixed 180 L with a variable.
- p people, 45 litres each a day.
- D = 14 137 ÷ 45p.
| Use per person | 45 L a day |
| People | p |
| Daily use | 45p litres |
| Days | D = 14 137 ÷ 45p |
Using the rule to predict
- Four people: 78 full days, as before.
- Six people over the holidays: 52 days.
- Two visitors cost 26 days of water.
| People | Working | Lasts |
|---|---|---|
| p = 4 | 14 137 ÷ 180 = 78.5 | 78 days |
| p = 6 | 14 137 ÷ 270 = 52.4 | 52 days |
| So | two visitors cost | 26 days |
A limitation: overflow
- The storm model assumed an empty tank.
- At 90% full, there is room for 1414 L.
- The rest of the storm spills.
| Value | ||
|---|---|---|
| Tank | 90% full | |
| Space left | 10% of 14 137 | 1414 L |
| Storm gives | 2400 L | |
| Kept | 1414 L | the rest spills |
What the model cannot tell you
- It ignores rain that falls while the tank empties.
- It treats daily use as fixed.
- Say which way each one moves the answer.
| Ignored | Effect |
|---|---|
| Rain during use | model says 78 days; really longer |
| Changing use | summer use is higher |
| Leaks, evaporation | small, but not zero |
| Sloped roof | plan area is what counts |
Say which way the answer moves
- Losses: more rain needed.
- More people: fewer days.
- Rain during use: more days.
When the model stops being useful
- Fine for a rough plan for a dry spell.
- Not for a drought lasting months.
- Say where the line is, and why.
An Excellence paragraph, spoken
- The answer, and the rule behind it.
- One assumption tested, with numbers.
- A limitation, and what it means.
What to take away
- An answer, and an honest account of how far to trust it.
| Extend | at least one problem, further |
| Assumptions | name them, then test one |
| Sensitivity | how far does the answer move? |
| General rule | replace a number with a variable |
| Predict | use the rule on a new case |
| Limitations | what the model ignores |
| Conclude | the answer and how far to trust it |
Your turn: four questions
- The same tank and roof.
- Pause here and answer each one.
- Question 4 wants a direction, not just a name.
| Question | |
|---|---|
| Tank | 14 137 L tank, 96 m² roof. |
| 1 | Only 75% of rain is collected. How much rain fills the empty tank? |
| 2 | Each person uses 50 L a day. Write D, the days a full tank lasts, for p people. |
| 3 | Use your rule to find D for 5 people. |
| 4 | Name one assumption in question 3 and how breaking it changes D. |
Show answer
1. 0.75 × 96 = 72, so 72d = 14.137 and d = 0.196 m: 196 mm of rain. 2. D = 14 137 ÷ 50p. 3. 14 137 ÷ 250 = 56.5, so 56 full days. 4. For example: use stays at exactly 50 L a person every day. If use rises on hot days, the tank empties sooner and D is smaller. Or: no rain falls; any rain makes D larger.
The answers
- 72d = 14.137, d = 0.196 m: 196 mm.
- D = 14 137 ÷ 50p.
- 14 137 ÷ 250 = 56.5: 56 full days.
- e.g. use is fixed; more use means fewer days.
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