Project Grade 12 Math
Sandalford's New Water Source: Desalination, Reuse, or Conservation
A grade 12 project where teams test desalination, reuse, conservation and a hybrid against supply and budget inequalities, then recommend one to a board.
Objectives
- Students will calculate annual volume, cost per acre-foot, annual operating cost, and annual energy use for four water-supply options, using weighted averages for blended options.
- Students will write supply and budget constraints as inequalities, identify which options are viable, and solve for the range of viable sizes of a hybrid option.
- Students will recommend one viable option to a board and state the tradeoff the recommendation accepts.
Vocabulary
- Desalination
- Turning salty water, like ocean water, into fresh water people can use. It works, but it takes a lot of energy and leaves behind very salty water that must be handled carefully.
- Potable reuse
- Cleaning used water so thoroughly that it becomes safe drinking water again. Special plants use several layers of treatment and constant testing to make sure it is safe.
- Acre-foot
- The amount of water it takes to cover one acre of land, about the size of a football field, one foot deep. Western states use acre-feet to measure large amounts of water.
Also: cost per acre-foot, capital cost, firm yield
Teacher overview
A coastal district facing a supply gap rarely has one obvious answer, because desalination, potable reuse, and conservation deliver water at very different prices, energy costs, and levels of drought reliability. Over five sessions, teams analyze the options on a price sheet for the fictional Sandalford Coastal Water District, test each against a drought-year supply requirement and a capital budget written as inequalities, and recommend one to the district's board. Two of the single options fail a constraint, so the decision comes down to a large reuse plant or a hybrid of conservation, brackish groundwater, and a small reuse plant, and teams must say which tradeoff they accept. The final deliverable is a written recommendation memo and a short presentation.
Project Brief
Driving question: Which new supply option should the fictional Sandalford Coastal Water District build to close a 5 MGD gap, given that the new supply must hold up in a drought year and stay within the capital budget, while keeping cost per acre-foot and energy use defensible?
Scenario and client: The Sandalford Coastal Water District (a fictional district) must add 5 MGD of firm new supply. The board plans for a declared drought year, so an option's firm yield is the amount it can still deliver in such a year. The board has asked student engineering teams to analyze the options below and recommend one, with numbers the board can defend to ratepayers.
Option price sheet (fictional figures for this project):
| Option | Normal-year yield | Drought-year (firm) yield | Capital cost | Operating cost | Energy |
|---|---|---|---|---|---|
| A. Seawater desalination | 5.0 MGD | 5.0 MGD | $150 million | $1,700 per acre-foot | 13 kWh per 1,000 gallons |
| B. Advanced potable reuse | 5.0 MGD | 5.0 MGD | $115 million | $1,300 per acre-foot | 4.0 kWh per 1,000 gallons |
| C. Conservation plus brackish groundwater | 5.0 MGD | 4.0 MGD | $60 million | see components below | see components below |
Notes on the options:
- Option A draws from the ocean, so its yield does not change in a drought.
- Option B treats water from the district's wastewater plant, which treats about 8 MGD in a normal year and about 7 MGD in a drought year. Either amount is enough to supply a 5 MGD reuse plant. The reuse plant can be built in any size from 1.0 to 5.0 MGD in steps of 0.5 MGD, and its capital cost is $25 million plus $18 million for each MGD of capacity. A 5.0 MGD plant is Option B.
- Option C has two components that are always bought together. The conservation program (fixture rebates, turf replacement, and leak repair) costs $20 million in capital, saves 2.0 MGD in every year, costs $500 per acre-foot saved to run, and uses no treatment energy. The brackish groundwater wells and treatment plant cost $40 million in capital, can pump 3.0 MGD in a normal year, cost $1,100 per acre-foot to run, and use 3.0 kWh per 1,000 gallons. The district's groundwater permit limits brackish pumping to 2.0 MGD in a declared drought year, which is why Option C's firm yield is 4.0 MGD.
- Hybrid option H: the board will also consider Option C combined with a reuse plant of r MGD, where r follows the size steps and cost rule for Option B. In the hybrid, the reuse plant runs at full capacity all year, conservation saves 2.0 MGD, and the brackish wells pump only what is needed to reach 5.0 MGD, up to the permit limit.
Constraints:
- Firm yield: the option's drought-year yield must be at least 5.0 MGD.
- Capital cost: the option's total capital cost may not exceed $120 million.
- The recommendation must report cost per acre-foot, annual operating cost, and annual energy use for the chosen option in a normal year, when it delivers exactly 5.0 MGD.
Useful references: One acre-foot equals 325,851 gallons. For scale, the real Carlsbad seawater plant produces about 50 MGD, and Orange County's reuse system produces about 130 MGD. Seawater reverse osmosis typically uses about 11 to 15 kWh per 1,000 gallons (about 3 to 4 kWh per cubic meter), compared with about 1 to 2 kWh per 1,000 gallons for conventional treatment and delivery of local surface water or groundwater.
Final deliverable: A one-to-two-page recommendation memo to the board plus a short team presentation, each stating the chosen option (and, for the hybrid, the reuse plant size), its cost per acre-foot, its annual operating cost, its annual energy use, the constraint check for every option, and the tradeoff the team accepted in choosing it over the other viable option.
Milestones
- Session 1, frame the problem: Teams restate the supply gap and the options in their own words, convert 5 MGD into an annual volume in gallons and in acre-feet, convert 1.0, 1.5, 2.0 and 3.0 MGD the same way for use with the component options, and list the constraints. Output: a one-page problem statement with the annual volumes computed.
- Session 2, cost per acre-foot: Teams calculate each option's normal-year cost per acre-foot and annual operating cost at 5.0 MGD. For Option C and for the hybrid at r = 1.0 and r = 1.5, they compute a weighted average cost per acre-foot from the components. Output: a cost table for Options A, B, C and the two hybrid sizes.
- Session 3, energy and reliability: Teams calculate the normal-year energy use of Options A, B and C and the two hybrid sizes from the kWh-per-1,000-gallons figures (conservation adds none) and record each option's drought-year yield from the price sheet notes. Output: an energy table and a short reliability note per option naming what its firm yield depends on.
- Session 4, write the constraints and choose: Teams write the firm-yield and capital constraints as inequalities, test Options A, B and C, and mark each viable or non-viable. They then write the hybrid's firm yield and capital cost as expressions in r, solve the resulting system of inequalities, and state which reuse plant sizes are viable under the 0.5 MGD step rule. Teams choose one viable option and name the tradeoff it accepts. Output: a constraints table, the solved hybrid system, and a stated recommendation.
- Session 5, write and present: Teams write the recommendation memo and present it to the class acting as the district board, defending the numbers and answering at least one board question about drought reliability or ratepayer cost. Output: the completed memo 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 |
|---|---|---|---|---|
| Quantitative analysis | Cost or energy figures are missing or incorrectly computed | Some figures are correct, but units are inconsistent or the blended options use a simple average instead of a weighted one | Annual volume, cost per acre-foot, annual operating cost, and annual energy are correct for every option, with weighted averages for the blended options and consistent units | All figures are correct and the team also shows a full cost per acre-foot that includes a 30-year share of capital cost |
| Meeting constraints | The recommendation ignores the firm-yield requirement or the $120 million cap | One constraint is checked, or both are checked for the single options but the hybrid is not analyzed | Both constraints are written as inequalities, Options A and C are shown non-viable and B viable, and the hybrid system is solved to give the viable reuse sizes | All of that, and the memo explains the tradeoff the team accepted, such as paying more each year for a reuse plant that does not depend on conservation, or accepting little or no drought margin to lower cost |
| Recommendation and communication | The memo states a choice with little or no supporting math | The memo gives a choice with partial support, or recommends a non-viable option | The memo states a viable choice, its cost per acre-foot, annual operating cost, and energy use, with clear reasoning | The memo does all that and anticipates a likely board question, such as what happens if conservation savings fall short or the groundwater permit is tightened |
Teacher key
Figures are computed from unrounded volumes and then rounded for reporting. Teams that round the volume first may differ by a few hundred dollars in annual cost; accept these.
Annual volumes (Milestone 1). 5 MGD x 365 days = 1,825,000,000 gallons per year, and 1,825,000,000 ÷ 325,851 = 5,600.7 acre-feet per year. For the components, 1.0 MGD = 1,120.1 acre-feet, 1.5 MGD = 1,680.2 acre-feet, 2.0 MGD = 2,240.3 acre-feet, and 3.0 MGD = 3,360.4 acre-feet per year.
Normal-year cost (Milestone 2). In the hybrid, the brackish wells pump 3.0 - r MGD in a normal year.
| Option | Normal-year mix | Cost per acre-foot | Annual operating cost |
|---|---|---|---|
| A | 5.0 MGD desalination | $1,700 | $9.52 million |
| B | 5.0 MGD reuse | $1,300 | $7.28 million |
| C | 2.0 conservation + 3.0 brackish | (2.0 x 500 + 3.0 x 1,100) ÷ 5.0 = $860 | $4.82 million |
| H, r = 1.0 | 2.0 conservation + 1.0 reuse + 2.0 brackish | (2.0 x 500 + 1.0 x 1,300 + 2.0 x 1,100) ÷ 5.0 = $900 | $5.04 million |
| H, r = 1.5 | 2.0 conservation + 1.5 reuse + 1.5 brackish | (2.0 x 500 + 1.5 x 1,300 + 1.5 x 1,100) ÷ 5.0 = $920 | $5.15 million |
A simple average of the component prices (for example, ($500 + $1,100) ÷ 2 = $800 for Option C) is the most common error and should be marked as a weighting mistake.
Normal-year energy (Milestone 3). One MGD for a year is 365,000 thousand gallons. Option A: 1,825,000 x 13 = 23,725,000 kWh. Option B: 1,825,000 x 4.0 = 7,300,000 kWh. Option C: brackish 3.0 MGD is 1,095,000 thousand gallons, and 1,095,000 x 3.0 = 3,285,000 kWh. Hybrid r = 1.0: 365,000 x 4.0 + 730,000 x 3.0 = 1,460,000 + 2,190,000 = 3,650,000 kWh. Hybrid r = 1.5: 547,500 x 4.0 + 547,500 x 3.0 = 2,190,000 + 1,642,500 = 3,832,500 kWh. Option A uses about 3.25 times the energy of Option B and about 6.5 times that of the hybrid at r = 1.0.
Constraints check (Milestone 4). With F for firm yield in MGD and K for capital cost in millions of dollars, the constraints are F ≥ 5 and K ≤ 120.
| Option | F ≥ 5? | K ≤ 120? | Result |
|---|---|---|---|
| A | 5.0 ≥ 5, true | 150 ≤ 120, false | Non-viable (over the cap by $30 million) |
| B | 5.0 ≥ 5, true | 115 ≤ 120, true | Viable ($5 million under the cap, no drought margin beyond 5.0 MGD) |
| C | 4.0 ≥ 5, false | 60 ≤ 120, true | Non-viable (1.0 MGD short in a drought year) |
For the hybrid, the drought-year yield is 2.0 (conservation) + 2.0 (brackish at the permit limit) + r, so 4 + r ≥ 5 gives r ≥ 1. The capital cost is 60 + 25 + 18r = 85 + 18r, so 85 + 18r ≤ 120 gives r ≤ 35/18, or about 1.94. The solution set is 1 ≤ r ≤ 1.94, and under the 0.5 MGD step rule the viable sizes are r = 1.0 (capital $103 million, drought-year yield exactly 5.0 MGD) and r = 1.5 (capital $112 million, drought-year yield 5.5 MGD, a 0.5 MGD margin). A 2.0 MGD reuse plant would cost $121 million and is non-viable.
Acceptable recommendations. Option B and the hybrid at either viable size are all defensible, and the grade should rest on the reasoning, not the choice. A team choosing B accepts about $2.2 million more per year in operating cost and about twice the energy of the hybrid in exchange for one drought-proof source that does not depend on customers keeping their conservation savings or on the groundwater permit. A team choosing the hybrid accepts that its firm yield relies on 2.0 MGD of conservation holding in a drought and on the permit not being tightened; choosing r = 1.5 over r = 1.0 pays $9 million more in capital and about $110,000 more per year for a 0.5 MGD drought margin. A memo that recommends Option A or Option C alone has not applied the constraints and cannot score above Developing on the Recommendation and communication row.
Full cost with capital (Advanced). Spreading capital evenly over 30 years of 5,600.7 acre-feet (168,021.6 acre-feet in all), with no interest, adds about $893 per acre-foot for A, $684 for B, $613 for the hybrid at r = 1.0, and $667 for the hybrid at r = 1.5. Full costs are then about $2,593, $1,984, $1,513, and $1,587 per acre-foot. Including capital does not change the ranking among the viable options in this case, yet it widens the gap between Option B and the hybrid from $400 to about $471 per acre-foot at r = 1.0.
Differentiation
Support
Provide the annual-volume conversion already worked out (5 MGD is 5,600.7 acre-feet per year) so teams can move straight to the cost table. Model the weighted-average method with an example that is not on the price sheet: a mix of 4.0 MGD at $600 per acre-foot and 1.0 MGD at $1,400 per acre-foot averages (4.0 x 600 + 1.0 x 1,400) ÷ 5.0 = $760 per acre-foot. For Milestone 4, give teams the two hybrid expressions, 4 + r for firm yield and 85 + 18r for capital cost, and ask them to write and solve the inequalities.
Extension
Ask teams to fold a 30-year share of each option's capital cost into a full cost per acre-foot and discuss whether including capital changes the comparison. Then pose a sensitivity question for the fictional district: if a new groundwater rule cut drought-year brackish pumping to 1.5 MGD, what would the hybrid's viable reuse sizes become, and would the team's recommendation change? (The firm-yield inequality becomes 3.5 + r ≥ 5, so r ≥ 1.5, and only r = 1.5 remains viable.)
ELL supports
Pre-teach "desalination," "reuse," "acre-foot," "capital cost," and "firm yield" with labeled visuals of each supply option. Provide a memo template with headed sections (Recommendation, Cost, Energy, Constraints, Tradeoff) and sentence frames for each, such as "Option ___ is non-viable because its ___ is ___, which does not meet the requirement of ___."
Extensions
Invite a local water-agency planner to review the teams' recommendations, or have teams compare their fictional Sandalford analysis with a real agency's published supply plan, noting how the real agency weighed cost, energy, and drought reliability.