Worksheet Grade 12 Engineering & Design

Nashmere High School Cistern Sizing and Payback Worksheet

A grade 12 worksheet sizing a rainwater cistern for a fictional school roof, from design-storm runoff and first flush to tank depth and simple payback.

Grade: Grade 12
Subject: Engineering & Design
Material: Worksheet
Time: 45 min
Prep: No prep
Materials: Calculator

Standards

HSG-MG.A.3 Common Core Math

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost).

How this material aligns: Items 4 through 6 use cylinder area and volume to set the depth of a fixed-diameter cistern so it satisfies a stated storage constraint drawn from the design storm.

All materials for HSG-MG.A.3 →

HSN-Q.A.1 Common Core Math

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How this material aligns: Items 1 through 12 carry square feet and inches of rain through gallons, gallons per season, dollars per thousand gallons, and years of payback in one consistent unit chain.

All materials for HSN-Q.A.1 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-09-03.

Objectives

Students will calculate roof runoff from a design storm using the rainfall-to-gallons conversion.
Students will size a fixed-diameter cistern by solving for the depth that meets a storage volume.
Students will compare a season's harvested supply against irrigation demand and find a simple payback.

Vocabulary

First flush: First flush is the very first bit of runoff at the start of a rainstorm, which often carries the most pollution because it washes off everything that built up on streets and roofs during the dry period before the rain. Later rainfall in the same storm is usually cleaner.
Cistern: A large tank that stores rainwater collected from a roof so it can be used later, usually for watering plants.

Also: design storm, capture efficiency, simple payback

Teacher overview

The fictional Nashmere High School wants to catch rain from its greenhouse roof and use it on the athletic field instead of buying treated drinking water for irrigation. This worksheet takes students through the sizing sequence a designer actually follows, starting with the runoff a single design storm produces, subtracting the first flush that gets diverted, solving for the tank depth that holds the rest, and only then checking the season's supply against the field's demand and the payback on the tank. The misconception this worksheet is built to correct is that annual rainfall sets the tank size. Annual rainfall tells a designer how much water the roof can supply over a year, while the tank only has to hold what arrives between uses, so sizing from the annual figure produces a tank many times larger and far more expensive than the site needs. Items 9 through 12 treat the whole year's harvest as usable. Because the cistern holds only about 13,500 gallons, most rain that falls outside the 22-week season overflows, so the real share of demand met and the annual saving are lower and the payback is longer. Ask students to name this assumption after item 12.

Nashmere High School Cistern Sizing and Payback Worksheet — Grade 12 Engineering & Design

Name: ____________________________ Date: ______________

Student Worksheet

The fictional Nashmere High School greenhouse roof measures 18,400 square feet. The design storm for the site is 1.2 inches of rain. Roof runoff in gallons equals roof area in square feet times rainfall in inches times 0.623. This is the rain that falls on the roof; a real system collects somewhat less, so a cistern sized from this figure has some room to spare. The cistern will be a vertical cylinder 24.0 feet in diameter, and cylinder volume in cubic feet equals 0.785 times the diameter squared times the depth. One cubic foot holds 7.48 gallons.

The site averages 34 inches of rain per year, and the system captures 85 percent of what falls on the roof over a year. The athletic field covers 43,560 square feet and needs 1.0 inch of water per week for a 22-week season. Treated water costs $5.40 per 1,000 gallons, and the installed cistern costs $2.10 per gallon of storage.

The greenhouse roof is 18,400 square feet and the design storm is 1.2 inches. Find the gallons of runoff the design storm produces.
A first-flush diverter sends the first 0.02 inch of rainfall to waste. Using a roof area of 18,400 square feet, find the gallons diverted.
Using a design storm runoff of 13,755.8 gallons and a first flush of 229.3 gallons, find the gallons the cistern must store.
The cistern is 24.0 feet in diameter. Using area equals 0.785 times the diameter squared, find its cross-sectional area.
Using a required storage of 13,526.5 gallons and 7.48 gallons per cubic foot, find the storage volume in cubic feet.
Using a required volume of 1,808.4 cubic feet and a cross-sectional area of 452.16 square feet, find the cistern depth in feet.
The roof is 18,400 square feet, the site receives 34 inches of rain per year, and the system captures 85 percent of it. Find the gallons harvested per year.
The athletic field is 43,560 square feet and needs 1.0 inch of water per week for 22 weeks. Find the season's irrigation demand in gallons.
Using an annual harvest of 331,286.5 gallons and a season demand of 597,033.4 gallons, find the percent of the season's demand the harvest can meet.
The installed cistern costs $2.10 per gallon of storage. Using a storage requirement of 13,526.5 gallons, find the installed cost of the cistern.
Treated water costs $5.40 per 1,000 gallons. Using an annual harvest of 331,286.5 gallons, find the annual value of the water the system replaces.
Using an installed cost of $28,405.65 and an annual saving of $1,788.95, find the simple payback in years, to the nearest tenth.
Explain in two or three sentences why the design storm, rather than the 34 inches of annual rainfall, sets the size of the cistern.

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

Answer Key

Answer: 13,755.8 gallons. At the fictional school, 18,400 square feet times 1.2 inches equals 22,080, and 22,080 times 0.623 equals 13,755.84 gallons.
Answer: 229.3 gallons. 18,400 square feet times 0.02 inch equals 368, and 368 times 0.623 equals 229.264 gallons.
Answer: 13,526.5 gallons. 13,755.8 gallons of runoff minus 229.3 gallons diverted equals 13,526.5 gallons to store.
Answer: 452.16 sq ft. 0.785 times 24.0 squared equals 0.785 times 576, which is 452.16 square feet.
Answer: 1,808.4 cubic feet. 13,526.5 gallons divided by 7.48 gallons per cubic foot equals 1,808.36 cubic feet.
Answer: 4.00 feet. 1,808.4 cubic feet divided by 452.16 square feet equals 3.9995 feet, which rounds to 4.00 feet of depth.
Answer: 331,286.5 gallons per year. 18,400 times 34 equals 625,600, and 625,600 times 0.623 equals 389,748.8 gallons. Multiplying by the 0.85 capture efficiency gives 331,286.48 gallons per year.
Answer: 597,033.4 gallons. 43,560 square feet times 1.0 inch times 0.623 equals 27,137.88 gallons per week, and 27,137.88 times 22 weeks equals 597,033.4 gallons for the season. Accept any value within a few gallons of this, since the weekly figure may be rounded first.
Answer: 55.5 percent. 331,286.5 divided by 597,033.4 equals 0.5549, which is 55.5 percent of the season's demand.
Answer: $28,405.65. 13,526.5 gallons times $2.10 per gallon of storage equals $28,405.65 installed.
Answer: $1,788.95 per year. 331,286.5 gallons is 331.2865 thousand gallons, and 331.2865 times $5.40 equals $1,788.95.
Answer: 15.9 years. $28,405.65 divided by $1,788.95 per year equals 15.88, which rounds to 15.9 years.
Answer: The tank only holds water between rain and use. Accept any answer explaining that the design storm sets the largest volume arriving at once, while the 34 inches of annual rainfall arrives across many separate storms and is drawn down by irrigation between them, so sizing to the annual total would buy storage that never fills at one time.

Differentiation

Support

Give these students items 1, 2, and 7 with the multiplication already arranged in order, so the runoff conversion becomes a sequence of keystrokes rather than a formula to reconstruct. Provide a worked model on a different fictional roof, such as 5,000 square feet catching a 1.0-inch storm, before item 1.

Extension

Ask these students to test a second geometry at the fictional school. Have them find the diameter a 6.0-foot-deep cistern would need to hold the same 13,526.5 gallons, then state which of the two tanks they would recommend for a courtyard with limited floor space and why.

ELL supports

Provide a labeled sketch of the roof, gutter, first-flush diverter, and cistern with an arrow showing the water path, so "first flush" and "storage" are distinct places on the drawing. Offer sentence frames for item 13: "The cistern must hold the water from ___," and "Annual rainfall arrives in ___, so it does not set ___."

Extensions

Suppose the fictional Nashmere district asks whether a second cistern would be worth building, since the harvest meets only part of the season demand. Ask students to calculate how much more of the season's demand could be met if the roof of the adjoining gymnasium, 26,000 square feet, were added to the collection system at the same capture efficiency.