Math problem set Grade 11 Math
Sarnford's Day of Demand: Sizing the Storage Tank
A grade 11 problem set building a running tank balance from a fictional district's 24-hour demand table, then sizing the tank in gallons and in feet.
Objectives
- Students will build a running balance of supply against demand across a 24-hour day.
- Students will determine the equalizing storage a distribution system needs from that running balance.
- Students will convert a required storage volume into a water depth in a cylindrical tank.
Vocabulary
Key words: equalizing storage, peak demand, running balance, fire reserve, cylindrical tank
Teacher overview
Water systems pump at a steady rate while customers use water in bursts, and the storage tank absorbs the difference, filling overnight and draining during the morning and evening peaks. This problem set gives grade 11 students a day of demand for the fictional Sarnford Water District in three-hour blocks and asks them to build the running balance that shows how the tank rises and falls, then size the tank from it. The error the set is built to catch is sizing the tank from the single largest block of demand, which ignores that the tank has already been drawn down before that block starts, and that the surplus stored earlier in the day is part of the answer.
Problems
The Sarnford Water District, its demand record, and its fire reserve requirement are fictional. Use these values throughout.
| Block | Hours | Demand (gallons) |
|---|---|---|
| 1 | midnight to 3 a.m. | 90,000 |
| 2 | 3 a.m. to 6 a.m. | 105,000 |
| 3 | 6 a.m. to 9 a.m. | 240,000 |
| 4 | 9 a.m. to noon | 210,000 |
| 5 | noon to 3 p.m. | 180,000 |
| 6 | 3 p.m. to 6 p.m. | 195,000 |
| 7 | 6 p.m. to 9 p.m. | 285,000 |
| 8 | 9 p.m. to midnight | 135,000 |
- The pumps run at one steady rate for all 24 hours and deliver exactly the day's total demand.
- The tank starts and ends the day at the same level.
- The storage tank is a cylinder 60 ft in diameter.
- Cylinder volume in cubic feet = 0.785 x diameter squared x depth, and 1 cubic foot = 7.48 gallons.
- The district must also hold a fire reserve of 1,500 gpm for 2 hours.
- Find the total demand for the day in gallons.
- Find the steady pumping rate in gallons per minute.
- How many gallons do the pumps deliver during one three-hour block?
- Build a running balance of pump delivery minus demand, block by block. What is the greatest number of gallons the tank holds above its starting level at any point in the day?
- From that same running balance, what is the greatest number of gallons the tank falls below its starting level?
- Find the equalizing storage the tank must provide, in gallons.
- Find the average flow to customers during block 7, in gallons per minute.
- Find the water depth, in feet, that the equalizing storage occupies in the 60 ft tank. Round to one decimal place.
- Find the depth needed for the equalizing storage and the fire reserve together. Round to one decimal place.
- A planner proposes sizing the tank at 285,000 gallons because that is the largest block of demand. Explain why the running balance gives a different and better answer.
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: 1,440,000 gal.
Adding the eight blocks: 90,000 + 105,000 + 240,000 + 210,000 + 180,000 + 195,000 + 285,000 + 135,000 = 1,440,000 gallons. - Answer: 1,000 gpm.
A day holds 24 x 60 = 1,440 minutes, and 1,440,000 gallons divided by 1,440 minutes = 1,000 gallons per minute. - Answer: 180,000 gal.
Three hours is 180 minutes, and 1,000 gallons per minute x 180 minutes = 180,000 gallons. The same figure comes from dividing the daily total by the eight equal blocks. - Answer: 165,000 gal.
Subtract each block's demand from the 180,000 gallons delivered, then keep a running total. Block 1 adds 90,000 for a running total of 90,000. Block 2 adds 75,000 for 165,000. Block 3 removes 60,000 for 105,000. Block 4 removes 30,000 for 75,000. Block 5 changes nothing and stays at 75,000. Block 6 removes 15,000 for 60,000. Block 7 removes 105,000 for -45,000. Block 8 adds 45,000 and returns the tank to 0. The highest running total is 165,000 gallons, reached at 6 a.m. - Answer: 45,000 gal.
The lowest running total is -45,000 gallons, reached at 9 p.m. at the end of the evening peak. - Answer: 210,000 gal.
The tank must cover the full swing from its highest point to its lowest, so the equalizing storage is 165,000 + 45,000 = 210,000 gallons. - Answer: 1,583 gpm.
Customers draw 285,000 gallons in block 7 across 3 hours, which is 180 minutes, so the average flow is 285,000 divided by 180 = 1,583.3 gallons per minute. That is about 1.58 times the daily average found in problem 2. - Answer: 9.9 ft.
Convert first: 210,000 gallons divided by 7.48 gallons per cubic foot = 28,074.9 cubic feet. The tank's cross-section is 0.785 x 60 x 60 = 2,826 square feet. Dividing gives 28,074.9 divided by 2,826 = 9.93 feet. - Answer: 18.4 ft.
The fire reserve is 1,500 gallons per minute x 120 minutes = 180,000 gallons, so the tank must hold 210,000 + 180,000 = 390,000 gallons. That is 390,000 divided by 7.48 = 52,139 cubic feet, and 52,139 divided by 2,826 = 18.4498 feet, which rounds to 18.4 feet. - Answer: The peak block ignores the drawdown that has already happened before it starts.
Storage has to cover the whole distance between the fullest moment and the emptiest moment, not the size of one block. By 6 p.m. the running balance has already fallen from its 165,000 gallon high down to 60,000 gallons, so the evening peak draws the tank down a further 105,000 gallons from there rather than from a full tank. Sizing at 285,000 gallons also overstates the need in the other direction, since the pumps keep running during the peak and supply 180,000 gallons of it. The running balance answer of 210,000 gallons accounts for both effects at once.
Differentiation
Support
Provide the running-balance table with the first three rows filled in and the remaining rows left blank, so students continue a pattern instead of designing one. Work problems 1 through 3 together as a class, and give students a printed number line from -60,000 to 180,000 on which to mark each running total as they compute it.
Extension
Ask students to sketch the running balance as a graph with time on the horizontal axis, then describe where the graph increases, where it decreases, and what the two turning points mean for an operator watching a tank level gauge. A strong follow-up is to recompute the equalizing storage if the pumps run only 16 hours a day instead of 24, which changes the delivery per block and therefore the whole balance.
ELL supports
Pre-teach "demand," "running balance," "equalizing storage," and "fire reserve," pairing each with a simple drawing of a tank at three different levels. Provide a labeled two-column table with the headings "water in" and "water out" so students can sort the numbers before subtracting, and allow problem 10 to be answered with a marked-up copy of the running-balance table plus one sentence.
Extensions
Have students find the storage volume and the service population reported by a nearby water utility, compute storage per person in gallons, and compare that figure against the 210,000 gallons this fictional district needs for equalizing storage alone. Utilities usually carry considerably more than the equalizing figure, and asking students to account for the difference leads directly into fire flow, emergency reserve, and the practical value of a tank that can ride through a pump failure.