Model rainwater collection and storage

Rain falls vertically onto a sloping roof. To model the volume of rain landing on it, which area should be multiplied by the rainfall depth?

A 2,000 L tank starts with 400 L. A 100 m² roof delivers 80% of its rainfall to the tank. A 20 mm storm fills it exactly. What changes if rainfall doubles to 40 mm?

Change rainfall from 10 to 20 to 40 mm. Catchment area stays 100 m², collection efficiency stays 80%, capacity stays 2,000 L and initial water stays 400 L. Assume no water use during the storm. The stored-water bar includes the initial 400 L.

Which statement fits the model at all slider settings?

Formulate and check a model

Define catchment area A in m², rainfall d in mm and collection efficiency e as a decimal. Let R be delivered litres, I initial litres and C usable capacity in litres. One millimetre on one square metre is 0.001 m³, or 1 L.

1. 1Choose assumptions This model assumes uniform vertical rainfall, a fixed collection fraction and no water use during the storm
2. 2Conserve water Initial water plus delivered water equals stored water plus overflow
3. 3Evaluate the result Check units, capacity constraints and whether observations support the assumed collection fraction. A mismatch needs investigation, not an invented cause

Catchment area

\(\displaystyle 120\text{ m}^2\)

Rainfall

\(\displaystyle 15\text{ mm}\)

Collection fraction

\(\displaystyle 0.8\)

A roof has horizontal catchment area 120 m². Rainfall is 15 mm and estimated collection efficiency is 80%. How many litres does the model deliver?

Multiply horizontal area by rainfall in millimetres and by the collection fraction

Internal dimensions (m)

\(\displaystyle 1.5\times0.8\times1.2\)

The usable tank interior is a rectangular prism 1.5 m long, 0.8 m wide and 1.2 m deep. What is its capacity in litres?

Multiply the internal dimensions in metres, then multiply cubic metres by 1000

Initial water

\(\displaystyle 300\text{ L}\)

Delivered water

\(\displaystyle 1440\text{ L}\)

Capacity

\(\displaystyle 1440\text{ L}\)

The 1,440 L tank initially holds 300 L. The storm delivers another 1,440 L and nobody uses water during it. How many litres overflow?

The tank has only 1,140 L of spare space. Compare incoming water with that spare space

Base area

\(\displaystyle 0.8\times1.2=0.96\text{ m}^2\)

Required volume

\(\displaystyle 1500\text{ L}=1.5\text{ m}^3\)

A new rectangular tank has internal base dimensions 0.8 m by 1.2 m. It must hold at least 1,500 L. Usable depth is specified in whole centimetres. What is the smallest depth that meets this model’s requirement?

The exact threshold is 1.5 ÷ 0.96 metres. Convert to centimetres and round upwards

A model predicts 1,440 L delivered, but a measuring device records 1,200 L. There was no overflow or water use. Which conclusion is justified?

A cylindrical tank has internal radius 0.8 m and usable depth 1.5 m. A rough estimate using π ≈ 3 gives 2,880 L. A calculation using π ≈ 3.14 gives 3,014.4 L. The target is 3,000 L. What is the sensible conclusion?

Catchment, rainfall, efficiency

\(\displaystyle \begin{gathered}A=90,\quad d=20\\e=0.75\end{gathered}\)

Initial water and capacity (L)

\(\displaystyle I=500,\quad C=1600\)

Transfer the model: a 90 m² catchment receives 20 mm of rain with 75% collection efficiency. A tank of usable capacity 1,600 L starts with 500 L. Assume no water use. How much overflows?

First find delivered water, then compare initial plus delivered water with capacity

An open modelling task

A community garden needs at least 2,000 L of usable rainwater storage. The available floor space is 1.6 m by 1.2 m and the maximum height is 1.8 m. Compare one rectangular-prism design with one vertical-cylinder design. For an initial model, treat walls and fittings as negligible; identify what must be checked before a real installation

1. 1Formulate and estimate Choose dimensions that fit the space. State your variables, units and assumptions, and estimate each capacity before calculating it
2. 2Solve and compare Calculate the capacities and check both the volume target and space constraints. Recommend a design using the results, without inventing prices or material specifications
3. 3Evaluate and communicate Write a short recommendation explaining the capacity margin, limitations and measurements or supplier details needed to validate it. This open written response needs teacher or reviewer feedback; it is not automatically marked