Project Grade 12 Math
Ravelston Field Cistern: A Rainwater Harvesting Design Proposal
A four-session grade 12 project sizing a cistern for a fictional school field, using a budget inequality and cylinder geometry under height and footprint limits.
Objectives
- Students will calculate roof runoff volume and irrigation demand from stated areas and depths.
- Students will write and solve a budget inequality for the largest affordable cistern capacity.
- Students will select cylinder dimensions that satisfy stated height and footprint limits at least cost.
Vocabulary
- Cistern
- A large tank that stores rainwater collected from a roof so it can be used later, usually for watering plants.
Also: catchment area, design storm, budget constraint, footprint, viable solution
Teacher overview
The fictional Ravelston High School irrigates its athletic field with treated drinking water, and its facilities committee has asked whether roof runoff from the gymnasium could carry part of that load. Over four sessions, teams size a cistern that fits a fixed budget, a fixed pad area, and a height limit, then present a written proposal with a scaled drawing. Teachers should expect a workable design near 11,000 to 12,000 gallons, which is well short of both a full design storm and a full week of irrigation, and that shortfall is the point of the project rather than a flaw in it. The strongest proposals say plainly what share of demand the cistern covers and defend the choice anyway.
Project Brief
Driving question: What is the largest cistern the fictional Ravelston High School can install within its budget and site limits, and what share of the field's irrigation demand will it actually cover?
Client: The fictional Ravelston High School facilities committee, which will hear each team's proposal in the final session.
The site. The gymnasium roof measures 24,000 square feet and drains to one downspout at the corner nearest the field. The athletic field measures 78,000 square feet. The only place a tank may stand is a level gravel pad area 24 feet by 24 feet beside the gymnasium, where a concrete pad must be poured under the tank, and a district rule in this scenario caps any tank at 12 feet in height. Buried supply piping from the tank to the field irrigation header runs 120 feet.
Design requirements.
- Use roof area in square feet times rainfall in inches times 0.623 to find gallons of runoff. This is the rain that falls on the roof; a real collection system captures somewhat less.
- Size the tank as a vertical cylinder, with volume in cubic feet equal to 0.785 times diameter squared times height, and 7.48 gallons per cubic foot.
- The design storm for this scenario is 1.2 inches of rain.
- The field needs 0.6 inches of water per week during the growing season.
- The cistern must supply at least one third of one week's irrigation demand.
- The tank must fit entirely on the 24-foot by 24-foot pad and stand no more than 12 feet tall.
Price sheet for this scenario.
| Item | Price |
|---|---|
| Tank, 6,000 gallons or larger | $1.40 per gallon of capacity |
| Tank, under 6,000 gallons | $1.85 per gallon of capacity |
| Pump, filter, and controls | $3,200 flat |
| Buried supply piping | $6.50 per linear foot |
| Concrete pad under the tank | $14.00 per square foot of tank footprint |
Total budget: $22,000, covering every line above.
Final deliverable. Each team submits a two-page written proposal to the facilities committee and presents it in five minutes. The proposal must include the runoff and demand figures, the budget inequality and its solution, the chosen tank capacity and dimensions, a scaled site drawing showing the tank on the pad, a constraint table showing every limit met, and one honest paragraph on what the cistern will not do.
Milestones
Session 1, catchment and demand. Calculate the gallons of runoff the gymnasium roof produces in the 1.2-inch design storm, and the gallons the field needs for one week at 0.6 inches. Compute the one-third-of-demand target the cistern must meet. Output: a one-page water balance sheet showing all three figures and the gap between roof supply and field demand.
Session 2, cost model and feasible size. Write total cost as a function of tank capacity, including the per-gallon tank price, the flat pump price, the piping run, and the pad cost, which depends on the tank's footprint. Set that expression against the $22,000 budget as an inequality and solve for the largest capacity the budget allows. Output: an itemized cost model, the inequality, its solution, and a statement of one capacity that is viable and one that is not, with reasons.
Session 3, geometry and constraint check. Convert the chosen capacity into a diameter and height, testing candidate heights against the 12-foot cap and candidate diameters against the 24-foot pad. Explain in writing why the least expensive tank of a given capacity is the tallest one the rule permits. Output: a scaled site drawing of the tank on the pad, labeled with diameter and height, plus a constraint table listing each limit, the design value, and whether it is met.
Session 4, proposal and hearing. Write the two-page proposal and present it to the class acting as the facilities committee. Each team fields one question from the committee about a tradeoff it made. Output: the completed proposal, the five-minute presentation, and a one-paragraph written answer to the committee's question.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Rubric
| Criterion | Emerging | Developing | Proficient | Advanced |
|---|---|---|---|---|
| Water balance | Runoff or demand figures are missing or use the wrong areas | Both figures appear but one conversion is applied incorrectly | Runoff, weekly demand, and the one-third target are all computed correctly from the stated areas and depths | All figures are correct and the proposal states the supply gap as a percent of weekly demand |
| Cost model and budget | Costs are listed without a model, or the total exceeds $22,000 | A cost model exists but omits the pad or piping line | Total cost is at or under $22,000 with every price sheet line included and the budget written as an inequality | The inequality is solved for maximum capacity and the team names one viable and one non-viable capacity with reasons |
| Geometry and site limits | Tank dimensions are stated without volume or limit checks | Dimensions produce the stated capacity but one site limit is unchecked | Dimensions produce the stated capacity, fit the 24-foot pad, and stay within the 12-foot height cap, shown in a constraint table | The team also explains why the tallest permitted tank minimizes pad cost at a fixed capacity |
| Drawing and proposal | Drawing is unscaled or the proposal omits required elements | Drawing is scaled but labels or one proposal element are missing | Scaled drawing is fully labeled and the proposal covers every required element clearly | Proposal states plainly what the cistern will not cover and defends the design against that limitation |
Differentiation
Support
Give teams the fictional Ravelston runoff figure and weekly demand already computed so Session 1 becomes a check rather than a first derivation, and supply the pad-cost term of the model as a printed expression in terms of capacity. Provide two pre-drawn candidate tanks on grid paper so the geometry work in Session 3 begins from concrete options.
Extension
Ask teams to model the fictional cistern across a full growing season rather than one week. Given a stated month-by-month rainfall pattern, they track the tank filling and drawing down, then report how many weeks per season the cistern runs dry and what capacity would be needed to cut that number in half.
ELL supports
Pre-teach "catchment," "cistern," "footprint," and "constraint" with a labeled diagram of the fictional site showing the roof, the downspout, the pad, and the field. Provide a constraint table template with the limits already printed and blanks for the design values, and give proposal paragraph frames such as "Our cistern holds ___ gallons, which covers ___ percent of one week of irrigation because ___."
Extensions
Have teams take the proposal to a wider audience by comparing their cistern with a second option the fictional committee is weighing, which is replacing part of the field with drought-tolerant turf. Teams estimate the water each option saves per season and the cost per thousand gallons saved, then recommend one. The comparison moves the project from a sizing exercise to a spending decision, which is the form the question actually takes when a school board hears it.