Math problem set Grade 6 Engineering & Design

Harrowfield Well 3 Carbon Filter Sizing and Cost Problems

Ten problems in which students size carbon treatment vessels for a fictional town's PFAS-affected well, then work out media cost, yearly cost, and percent of budget.

Grade: Grade 6
Subject: Engineering & Design
Material: Math problem set
Time: 40 min
Prep: No prep
Materials: Calculator

Standards

MS-ETS1-1 NGSS

Define the criteria and constraints of a design problem with sufficient precision to ensure a successful solution, taking into account relevant scientific principles and potential impacts on people and the natural environment that may limit possible solutions.

How this material aligns: Problems 3 and 10 size the vessel set against the well's stated flow requirement, and problem 10 rejects an undersized vessel on that constraint before its price is considered.

All materials for MS-ETS1-1 →

6.RP.A.3 Common Core Math

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations, including finding a percent of a quantity and converting measurement units.

How this material aligns: Problem 9 asks what percent of the annual carbon set-aside the yearly media cost uses, a direct percent-of-a-quantity application on a rate-based scenario.

All materials for 6.RP.A.3 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will divide a required well flow by a vessel's rated capacity to find how many treatment vessels a design needs.
Students will calculate the purchase cost and the monthly cost of carbon media from a price per pound.
Students will find what percent of an annual budget the yearly carbon cost uses.

Vocabulary

Granular activated carbon: Small black grains of specially treated carbon that grab contaminants as water flows through them. Utilities use big tanks of it to remove PFAS, bad tastes, and odors.
Flow rate: Flow rate measures how much water moves past a point in a certain amount of time, like gallons per minute through a garden hose. Bigger pipes and stronger pushes give bigger flow rates.

Also: capacity, constraint, media

Teacher overview

The town of Harrowfield is fictional, and every flow, price, and budget figure in this set was invented for it. Sampling at Harrowfield's Well 3 found PFAS, a family of manufactured chemicals for which a 2024 federal rule sets a limit of 4.0 parts per trillion for PFOA and PFOS, with the deadline for meeting it still being settled, so the utility plans to add granular activated carbon vessels, which hold the chemicals on the surface of the carbon as water passes through. Students size the vessel set against the well's flow, then work out what the carbon costs to buy, to replace, and to carry in the annual budget. Students often divide the total flow by however many vessels a supplier happens to list. The set works against that habit, because the number of vessels a design needs comes from dividing the required flow by one vessel's rated capacity.

Harrowfield Well 3 Carbon Filter Sizing and Cost Problems — Grade 6 Engineering & Design

Name: ____________________________ Date: ______________

Problems

Every number below belongs to the fictional Harrowfield scenario.

Well 3 pumps 350 gallons each minute. How many gallons does it pump in one hour?
Harrowfield runs Well 3 for 16 hours a day. Using your answer from problem 1, how many gallons does the well pump in one day?
Each carbon vessel can treat 175 gallons per minute. How many vessels does the town need to treat all 350 gallons per minute?
Each vessel holds 4,000 pounds of carbon. How many pounds of carbon fill the vessels you found in problem 3?
Carbon costs $2.15 per pound. What does it cost to fill those vessels once?
The carbon has to be replaced every 20 months. Spread the cost from problem 5 evenly. What is the carbon cost for one month?
Harrowfield keeps replacing the carbon on that schedule year after year. Using your answer from problem 6, what is the carbon cost for 12 months?
The town sets aside $12,000 a year for carbon. How many dollars under that amount is your answer from problem 7?
What percent of the $12,000 set-aside does the yearly carbon cost use? Round to the nearest whole percent.
A supplier offers one large vessel rated for 300 gallons per minute in place of the smaller ones. Well 3 pumps 350 gallons per minute. Explain why that single vessel does not meet the design requirement.

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: 21,000 gallons. The well pumps 350 gallons each minute, and one hour holds 60 minutes, so 350 x 60 = 21,000 gallons.
Answer: 336,000 gallons. One hour gives 21,000 gallons, and the well runs 16 hours, so 21,000 x 16 = 336,000 gallons.
Answer: 2 vessels. Divide the required flow by each vessel's rated capacity: 350 / 175 = 2 vessels.
Answer: 8,000 pounds. Two vessels hold 4,000 pounds each, so 2 x 4,000 = 8,000 pounds.
Answer: $17,200. Carbon runs $2.15 a pound, so 8,000 x 2.15 = 17,200 dollars.
Answer: $860. One load costs 17,200 dollars and lasts 20 months, so 17,200 / 20 = 860 dollars a month.
Answer: $10,320. Twelve months at 860 dollars each gives 860 x 12 = 10,320 dollars.
Answer: $1,680. The set-aside is 12,000 dollars and the yearly cost is 10,320 dollars, so 12,000 - 10,320 = 1,680 dollars under.
Answer: 86 percent. Divide the yearly cost by the set-aside: 10,320 / 12,000 = 0.86, which is 86 percent.
Answer: The single vessel is undersized. It is rated for 300 gallons per minute, which is 50 gallons per minute short of the 350 gallons per minute Well 3 delivers, so some untreated water would have to bypass the vessel. A design has to meet the flow requirement before price is worth comparing.

Differentiation

Support

Give students a printed page from the fictional supplier's catalog that shows the two vessels drawn side by side with 175 gallons per minute printed under each one, so problem 3 can be solved by counting rather than by choosing an operation. Supply the conversion 1 hour = 60 minutes on the same page.

Extension

Ask students to find the price per pound at which the yearly carbon cost in this fictional scenario would use the entire 12,000 dollar set-aside, working backward from 12,000 dollars a year to a monthly figure, then to one 20-month load, then to a price per pound.

ELL supports

Provide a simple cross-section drawing of a treatment vessel with an arrow entering the top and leaving the bottom, labeled with the words "flow rate," "capacity," and "carbon." Allow students to answer problem 10 with a labeled sketch and two numbers rather than a written paragraph.

Extensions

Have students look up the water quality report their own utility publishes each year and list which contaminants it reports, without comparing any figure to the fictional Harrowfield numbers. Ask them to write two sentences on why a utility would report results even when every result is below the limit.