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Math problem set Grade 12 Careers & CTE

Skelwick Valley Reuse Program: Acre-Foot Cost Math

Grade 12 problem set on recycled water economics: annual production in acre-feet, revenue, avoided cost, payback, energy use, and a marginal-cost supply argument.

Objectives

  1. Students will convert an annual recycled water production volume from million gallons per day to acre-feet.
  2. Students will calculate program revenue, avoided customer cost, and simple payback on a pipeline investment.
  3. Students will compare two supply options using marginal cost per acre-foot and defend a recommendation.

Vocabulary

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.
Recycled water
Wastewater that has been cleaned and used again, often to water parks, golf courses, and crops instead of being thrown away. Purple pipes usually carry it.
Tertiary treatment
Tertiary treatment is an extra round of cleaning wastewater gets after primary and secondary treatment, aimed at removing pollutants that earlier steps could not catch, like extra nutrients or fine particles. It produces very clean water, sometimes clean enough to reuse.

Also: avoided cost, simple payback, marginal cost

Teacher overview

A reuse program coordinator at the fictional Skelwick Valley Water District sells treated recycled water to irrigation customers and has to justify the program to a board that reviews supply in acre-feet and dollars rather than in gallons. This set moves through one program year, from production volume to revenue, avoided customer cost, pipeline payback, treatment energy, and finally a marginal-cost comparison between expanding reuse and buying more imported water. The misconception the set corrects is the assumption that a price quoted per acre-foot can be compared directly with a price quoted per thousand gallons, since an acre-foot holds 325,851 gallons and the two prices differ by more than two orders of magnitude before conversion. Problem 7 makes that conversion explicit.

Skelwick Valley Reuse Program: Acre-Foot Cost Math — Grade 12 Careers & CTE
Name: ____________________________ Date: ______________

Problems

Use the following figures for the fictional Skelwick Valley Water District. The district's water reclamation plant delivers 3.2 MGD of recycled water to irrigation customers every day of the year. Recycled water is sold under contract at $940 per acre-foot, while the imported raw water those same customers would otherwise buy costs $1,480 per acre-foot. The district financed a distribution pipeline for the program at a capital cost of $9,600,000. The tertiary filtration and disinfection step at the plant uses 1.4 kWh per 1,000 gallons treated, and the district pays $0.11 per kilowatt-hour. Use 325,851 gallons per acre-foot.

  1. Calculate the district's annual recycled water production in gallons, treating the year as 365 days.
  2. Convert the annual production from problem 1 to acre-feet. Round to the nearest whole acre-foot.
  3. Calculate the district's annual revenue from recycled water sales at the contract price, using your rounded answer from problem 2.
  4. Calculate what those customers would have paid for the same volume of imported raw water at its stated price.
  5. Calculate the customers' total annual savings from buying recycled water instead of imported water.
  6. Calculate the simple payback period on the pipeline in years, dividing the capital cost by the annual customer savings from problem 5. Round to one decimal place.
  7. Convert the recycled water contract price to dollars per thousand gallons. Round to the nearest cent.
  8. Calculate the annual electricity used by the tertiary treatment step, in kilowatt-hours, using the annual production from problem 1.
  9. Calculate the annual cost of that electricity at the stated rate.
  10. The district can add 500 acre-feet per year of supply either by expanding the reuse plant at a marginal cost of $1,150 per acre-foot or by purchasing additional imported water at its stated price. In two or three sentences, argue for one option using the marginal cost per acre-foot of each, and name which option you recommend.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 1,168,000,000 gallons. Annual production = 3.2 million gallons per day x 365 days = 1,168 million gallons, which is 1,168,000,000 gallons.
  2. Answer: 3,584 acre-feet. Volume = 1,168,000,000 gallons / 325,851 gallons per acre-foot = 3,584.46, which rounds to 3,584 acre-feet. Students who multiply instead of dividing will report a volume far larger than any district could deliver, which is the signal that the conversion factor was applied in the wrong direction.
  3. Answer: $3,368,960. Revenue = 3,584 acre-feet x $940 per acre-foot. Breaking the multiplication apart, 3,584 x 900 = $3,225,600 and 3,584 x 40 = $143,360, for a total of $3,368,960.
  4. Answer: $5,304,320. Imported cost = 3,584 acre-feet x $1,480 per acre-foot. Breaking it apart, 3,584 x 1,000 = $3,584,000 and 3,584 x 480 = $1,720,320, for a total of $5,304,320.
  5. Answer: $1,935,360. Savings = $5,304,320 minus $3,368,960 = $1,935,360 per year across all program customers.
  6. Answer: 5.0 years. Payback = $9,600,000 / $1,935,360 per year = 4.96 years, which rounds to 5.0 years.
  7. Answer: $2.88 per thousand gallons. One acre-foot is 325,851 gallons, which is 325.851 thousand gallons. Dividing, $940 / 325.851 = $2.8847, which rounds to $2.88 per thousand gallons.
  8. Answer: 1,635,200 kWh. The annual production of 1,168,000,000 gallons is 1,168,000 thousand gallons. Energy = 1,168,000 x 1.4 kWh per thousand gallons = 1,635,200 kilowatt-hours.
  9. Answer: $179,872. Cost = 1,635,200 kWh x $0.11 per kWh = $179,872 per year.
  10. Answer: Expand the reuse plant. Expanded reuse costs $1,150 per acre-foot at the margin while imported water costs $1,480 per acre-foot, so each of the 500 acre-feet costs $330 less from the reuse plant. Across the full 500 acre-feet that difference is $165,000 per year, and a complete argument should also note that the reuse supply is produced locally and does not depend on an imported allocation that can be cut in a dry year.

Differentiation

Support

Give students the acre-foot conversion worked out for a single day of production before they attempt problem 1, so the annual multiplication is the only new step. Provide problems 3 through 5 as a partially completed table with column headings for volume, unit price, and total, and let students fill only the totals. Students who complete problems 1 through 6 have met the first two objectives.

Extension

Ask students to recalculate the payback period if the district could only sell three quarters of its production in the first two years, then explain how a slower ramp-up changes the case a coordinator would make to the board. Students can also determine what contract price would leave customers indifferent between recycled and imported water, and discuss why a district would price below that point.

ELL supports

Pre-teach acre-foot, recycled water, avoided cost, payback, and marginal cost with a one-page glossary that pairs each term with its unit. Provide a labeled diagram showing water moving from the reclamation plant through the pipeline to irrigation customers, so the pipeline investment in problem 6 has a physical referent. For problem 10, offer a sentence frame that begins with the recommendation and then asks for the two unit costs as evidence.

Extensions

Have students locate the recycled water rate schedule published by an actual water agency, convert every price on it to dollars per thousand gallons, and compare the resulting figures with the potable rates in the same schedule. Students can also interview a local utility about which customers are physically able to take recycled water, since a program's economics depend as much on pipeline routing as on price.