Project Grade 10 Math
Fernbrook High Rainwater Cistern: A Sizing Design Project
Grade 10 teams size a 3,000-gallon rainwater cistern for a fictional school, pricing the tank by its surface area to meet a $5,000 budget and a 12-foot pad.
Objectives
- Students will calculate roof runoff and cylinder volume in gallons from given dimensions.
- Students will calculate the surface area of a closed cylindrical tank and use it to price a design against a fixed budget.
- Students will design a cistern that meets a storage target within a budget and footprint limit and explain how tank shape affects cost.
Vocabulary
- Rainwater harvesting
- Collecting rain, usually from a roof, and storing it in barrels or tanks for later use, like watering gardens. It saves treated drinking water for the uses that really need it.
- Cistern
- A large tank that stores rainwater collected from a roof so it can be used later, usually for watering plants.
Also: roof runoff, surface area, budget constraint
Teacher overview
Rainwater harvesting captures roof runoff in a tank, or cistern, and holds it for later use such as summer irrigation. Sizing a cistern is a design problem, because the tank must hold enough water, fit the space, and stay within budget. In this project the tank is priced by the area of steel in its wall, floor, and lid, so two tanks of the same volume can cost different amounts. Over four sessions, teams size a cylindrical cistern for a fictional school, find a shape that meets every limit, and pitch it to a stand-in facilities committee. The final deliverable is a scaled drawing with an itemized cost and a one-page written proposal.
Project Brief
Driving question: How can we size a rainwater cistern that stores at least 3,000 gallons of roof runoff for summer irrigation, within a $5,000 budget and a 12-foot by 12-foot concrete pad?
Scenario: Fernbrook High School is a fictional school whose grounds crew wants to water the school's garden beds and landscaping from captured rain instead of city water. The crew has asked teams to size a cistern and propose it to the school's facilities committee, which is played by classmates. The concrete pad is already in place.
Given data (set for this project):
- Roof area draining to the cistern: 12,000 square feet
- Design storm: 1.5 inches of rain
- Roof-runoff formula: gallons = roof area (sq ft) x rainfall (inches) x 0.623, which gives the rain falling on the roof; a real system collects somewhat less, and installers often assume 75 to 90 percent
- Cylinder volume: cubic feet = 0.785 x diameter squared x depth; then multiply by 7.48 for gallons
- Treat the tank as full to its top, so the water depth equals the height of the tank wall
- Wall area of the tank (sq ft) = 3.14 x diameter x depth
- Floor area and lid area of the tank (sq ft) = 0.785 x diameter squared for each
Fernbrook price sheet (fictional prices set for this project):
| Item | Price |
|---|---|
| Tank shell of coated steel, covering the wall, floor, and lid | $10.00 per square foot of shell surface |
| First-flush diverter, which sends the dirtiest first rain away from the tank | $250 |
| Gutter connection and inlet piping | $400 |
| Overflow pipe with screened outlet | $200 |
| Outlet valve and hose fitting | $150 |
| Delivery and installation | $500 |
Every design must include all six lines. The five fixed items total $1,500, so the shell may cost no more than $3,500, which means the shell can have no more than 350 square feet of surface.
Constraints:
- The cistern must hold at least 3,000 gallons.
- It must sit on the 12-foot by 12-foot pad, so the diameter cannot exceed 12 feet.
- The total cost, priced from the Fernbrook price sheet, must stay at or under $5,000.
Final deliverable: a scaled drawing of the cistern on grid paper with its diameter, depth, volume, and shell area labeled, an itemized cost from the price sheet, and a one-page written proposal to the facilities committee showing that the design meets the volume, space, and budget limits.
Milestones
- Session 1, find the target: Teams calculate the runoff from a 1.5-inch storm on the 12,000-square-foot roof, then confirm that a 3,000-gallon cistern holds only part of it, even if the system collects just 75 percent of the rain. Teams also find the volume in cubic feet that 3,000 gallons requires. Output: a completed calculation sheet.
- Session 2, draft two cistern options: Teams choose two diameter-and-depth pairs that each hold at least 3,000 gallons and fit the 12-foot pad. The two options must differ in shape, with one wide and shallow (diameter 10 feet or more) and one narrow and deep (diameter 7 feet or less). Teams compute each volume in gallons. Output: two labeled sketches with volume work.
- Session 3, cost and choose one: Teams compute the wall, floor, and lid areas of each option, price each one line by line from the Fernbrook price sheet, and check it against the $5,000 budget. If neither option fits, teams try a third diameter between the two. The class then posts each team's diameter and shell area on a shared table, and every team writes two or three sentences explaining which shape of tank used the least steel and why. Teams pick one design that meets all three limits and refine it into a final scaled drawing. Output: one final drawing with an itemized cost at or under $5,000 and the written shape explanation.
- Session 4, write and present: Teams write the one-page proposal and present the drawing and proposal to the class acting as the facilities committee. Output: the written proposal and a short presentation.
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 |
|---|---|---|---|---|
| Runoff and volume math | Runoff or volume calculations are missing or use the wrong formula | Formulas are used but with unit slips or arithmetic errors | Runoff and cistern volume are computed correctly with units shown | Calculations are correct, clearly labeled, and checked a second way |
| Surface area and cost | Shell area or cost is missing, or prices are applied to volume instead of area | Shell area is computed but the wall, floor, or lid is left out, or a fixed line from the price sheet is missing | Wall, floor, and lid areas are correct and the itemized cost includes all six price sheet lines | Cost is correct and itemized, and the team compares at least three designs to explain why a tank whose depth is close to its diameter uses the least steel |
| Design within constraints | Design misses the volume, space, or budget limit | Design meets one or two limits but not all three | Design holds at least 3,000 gallons, has a diameter of 12 feet or less, and totals $5,000 or less on the price sheet | Design meets all three limits and explains a tradeoff the team chose |
| Proposal and presentation | Proposal is incomplete or hard to follow | Proposal states the design but gives little reasoning | Proposal clearly shows how each limit is met | Proposal shows each limit met and answers a likely committee question |
Teacher check values. The runoff from the design storm is 12,000 x 1.5 x 0.623 = 11,214 gallons, and at 75 percent collection the system would still capture about 8,411 gallons, so a 3,000-gallon cistern holds only part of it. Storing 3,000 gallons takes 3,000 / 7.48 = about 401.1 cubic feet. The table below shows sample designs, each using the shallowest depth, to the nearest tenth of a foot, that reaches 3,000 gallons. Areas are rounded to two decimal places, and costs are computed from the unrounded areas and rounded to the nearest dollar, so a team that prices the rounded 6-foot shell area will get $4,740.50.
| Diameter (ft) | Depth (ft) | Volume (gal) | Wall area (sq ft) | Floor plus lid (sq ft) | Shell area (sq ft) | Total cost | Within $5,000? |
|---|---|---|---|---|---|---|---|
| 5 | 20.5 | 3,009 | 321.85 | 39.25 | 361.10 | $5,111 | No |
| 6 | 14.2 | 3,002 | 267.53 | 56.52 | 324.05 | $4,740 | Yes |
| 8 | 8.0 | 3,006 | 200.96 | 100.48 | 301.44 | $4,514 | Yes |
| 10 | 5.2 | 3,053 | 163.28 | 157.00 | 320.28 | $4,703 | Yes |
| 12 | 3.6 | 3,044 | 135.65 | 226.08 | 361.73 | $5,117 | No |
A tank that uses the whole 12-foot pad goes over budget because its large floor and lid add more steel than the short wall saves, and a very narrow tank goes over budget because its tall wall adds more than the small floor and lid save. Diameters from about 5.3 to 11.6 feet can meet the budget at 3,000 gallons. The least steel for a given volume comes when the depth equals the diameter, which is the 8-foot by 8-foot design in the table. Teams do not need to reach that exact design to score Proficient, but their shape explanation in Session 3 should state that the cost falls as the shape moves from either extreme toward a depth close to the diameter.
Differentiation
Support
Give teams the Session 1 runoff calculation already worked, so they can focus on cistern geometry and cost. Provide the table below, which lists the gallons held by each foot of depth for several diameters (0.785 x diameter squared x 7.48), so teams can find a depth by dividing 3,000 by the number in the table. Provide a cost worksheet with blank lines for the wall, floor, and lid areas and for each of the six price sheet items.
| Diameter (ft) | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|
| Gallons per foot of depth | 211.4 | 287.7 | 375.8 | 475.6 | 587.2 | 710.5 | 845.5 |
Extension
Have teams write the shell area of a 3,000-gallon tank as a function of diameter alone. Because the depth must equal 401.1 / (0.785 x diameter squared), the wall area simplifies to 4 x 401.1 / diameter, so the shell area is about 1,604.4 / diameter + 1.57 x diameter squared. Teams graph this function for diameters from 4 to 12 feet, read the lowest point from the graph, and use the 350-square-foot shell limit to find the narrowest and widest diameters the budget allows. Teams can then propose a larger cistern that captures more of the 11,214 gallons and find the largest volume a tank whose depth equals its diameter could hold within the same fictional $5,000 budget. For the teacher, the graph's lowest point is near a diameter of 8 feet, at about 301 square feet of shell. The 350-square-foot limit allows diameters from about 5.2 to 11.6 feet when the depth is not rounded. A tank whose depth equals its diameter has a shell area of 4.71 x diameter squared, so within the limit it can be about 8.6 feet across and 8.6 feet deep and hold about 3,760 gallons.
ELL supports
Pre-teach "cistern," "runoff," "surface area," and "budget" with a labeled diagram showing rain on a roof flowing into a tank, with the tank's wall, floor, and lid shaded in different colors. Provide a proposal template with headings and sentence frames such as "Our cistern holds ___ gallons because " and "Our tank costs $, which is under the budget because ___."
Extensions
Teams can compare the cistern's summer irrigation supply against the garden's estimated weekly water need and report how many weeks of dry weather the full cistern could cover, turning the design into a simple water-budget check.