Worksheet Grade 9 Math
Ormsleigh School's Rain Cistern: Roof Yield and Tank Size
A grade 9 worksheet where students model a fictional school cistern as a cylinder, find the roof runoff from a storm, and size the tank to hold it.
Objectives
- Students will calculate roof runoff volume from a roof area and a rainfall depth.
- Students will model a storage tank as a cylinder and compute its capacity in gallons.
- Students will solve for a tank diameter that meets a stated storage requirement.
Vocabulary
- Cistern
- A large tank that stores rainwater collected from a roof so it can be used later, usually for watering plants.
Also: runoff, catchment, capacity
Teacher overview
Rain that lands on a roof runs off within minutes, and a cistern captures some of it for later use on a garden or a lawn. Sizing a cistern means comparing two numbers: how much runoff a storm delivers, and how much the tank can hold. This worksheet has students work both sides of that comparison for the fictional Ormsleigh School, then solve for the tank diameter that would hold a whole design storm. A frequent error in the tank calculation is substituting the radius into a formula written for the diameter, which makes the computed capacity four times too small.
Student Worksheet
Use these values for the fictional Ormsleigh School in every item.
- Roof: a rectangle 120 feet by 65 feet
- Design storm: 1.5 inches of rain
- Proposed cistern: a cylinder 12 feet in diameter and 8 feet deep
- Runoff conversion: roof area (sq ft) x rainfall (inches) x 0.623 = gallons of rain falling on the roof. A real system collects somewhat less; in these items, assume all of it reaches the cistern.
- Cylinder volume (cu ft) = 0.785 x diameter squared x depth
- 1 cubic foot = 7.48 gallons
- School garden demand: 250 gallons per day
- Find the area of the roof in square feet.
- Find the gallons of runoff the roof delivers during the 1.5-inch design storm.
- Model the cistern as a cylinder and find its volume in cubic feet.
- Convert the cistern volume from item 3 into gallons. Round to the nearest tenth.
- The cistern starts empty. Find how many gallons of the design storm's runoff would spill over once the tank is full. Round to the nearest tenth.
- Find the rainfall depth, in inches, that would exactly fill the empty cistern. Round to the nearest hundredth of an inch.
- Starting from a full cistern, find how many whole days the garden's 250-gallon daily demand can be met before the tank runs dry.
- The school wants a cistern that holds the entire design storm at the same 8-foot depth. Find the diameter it would need, to the nearest tenth of a foot.
- Ormsleigh averages 38 inches of rain in a year. Find the total annual roof runoff in gallons.
- The average US household uses about 300 gallons of water per day. Find how many days of that use your item 9 answer would cover. Round to the nearest whole day.
- Explain why raising the cistern diameter by 25 percent adds more capacity than raising the depth by 25 percent.
- Give one reason the school should size the cistern for a typical storm rather than for the largest storm on record.
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: 7,800 sq ft. Area = 120 x 65 = 7,800 square feet.
- Answer: 7,289.1 gallons. Runoff = 7,800 sq ft x 1.5 in x 0.623 = 11,700 x 0.623 = 7,289.1 gallons.
- Answer: 904.32 cu ft. Volume = 0.785 x 12 x 12 x 8 = 0.785 x 144 x 8 = 113.04 x 8 = 904.32 cubic feet. The formula uses the diameter, not the radius.
- Answer: 6,764.3 gallons. 904.32 cu ft x 7.48 gallons per cubic foot = 6,764.3136, which rounds to 6,764.3 gallons.
- Answer: 524.8 gallons. Runoff minus capacity = 7,289.1 - 6,764.3 = 524.8 gallons spill over.
- Answer: 1.39 inches. The roof yields 7,800 x 0.623 = 4,859.4 gallons per inch of rain. Then 6,764.3 / 4,859.4 = 1.3920, which rounds to 1.39 inches.
- Answer: 27 days. 6,764.3 / 250 = 27.06 days, and only whole days are fully supplied, so the answer is 27 days.
- Answer: 12.5 feet. The tank must hold 7,289.1 gallons, which is 7,289.1 / 7.48 = 974.5 cubic feet. Then 974.5 = 0.785 x d squared x 8, so d squared = 974.5 / 6.28 = 155.18 and d = 12.46, which rounds to 12.5 feet.
- Answer: 184,657.2 gallons. Annual runoff = 7,800 sq ft x 38 in x 0.623 = 296,400 x 0.623 = 184,657.2 gallons.
- Answer: 616 days. 184,657.2 / 300 = 615.52 days, which rounds to 616 days.
- Answer: Volume depends on the square of the diameter but only on the first power of the depth. Raising the diameter by 25 percent multiplies the capacity by 1.25 squared, which is 1.5625, while raising the depth by 25 percent multiplies it by 1.25. The wider tank gains about 56 percent capacity against about 25 percent for the deeper one.
- Answer: A tank sized for the record storm would be far larger, more expensive, and would sit mostly empty in a normal year. Sizing for a typical storm keeps the cost and the footprint reasonable, and the overflow from a rare storm can be routed to a garden bed or a storm drain.
Differentiation
Support
Give students the cylinder formula with the diameter already substituted for this fictional tank, so the first calculation is only the multiplication. Provide a labeled sketch of the roof and the cistern with each given measurement written on it, and work items 1 and 2 together before students continue.
Extension
Ask students to compare two tanks of equal capacity in the same fictional scenario, one short and wide and one tall and narrow, and calculate the surface area of the curved side of each. Have them explain which shape uses less material and why a school might still choose the other one.
ELL supports
Pre-teach "cistern," "runoff," and "capacity" with a picture card showing rain on a roof, a downspout, and a tank. Provide a sentence frame for item 12: "A tank built for the largest storm would ___, so the school should ___."