Math problem set Grade 10 Math
Bellamy Lead Service Line Replacement Problem Set
A grade 10 problem set on a fictional utility's lead line replacement, modeled as a linear function with a 10-year deadline, per-line cost, and a changing action level.
Objectives
- Students will model lead service line replacement as a linear function and solve for time or rate.
- Students will compute replacement cost and compare a test reading to a changing action level.
Vocabulary
- Lead service line
- A lead service line is an old water pipe made of lead metal that connects a house to the water main. Lead can get into the drinking water, so these pipes are being dug up and replaced across the country.
- Action level
- A trigger number in lead and copper testing. If too many homes test above this level, the utility must take steps like improving treatment and telling the public.
Also: linear model, replacement rate
Teacher overview
EPA estimates, from the service line inventories utilities reported in 2025, that about 4 million lead service lines still connect homes to water mains across the country, and federal rules now require utilities to replace nearly all of them within 10 years. The fictional Bellamy Water System uses that mandate to set a replacement schedule, which students model as a linear function and price out. A common misconception here is treating the replacement as exponential growth or decay; because Bellamy removes a fixed number of lines each year, the count of lines remaining drops by equal amounts over equal times, which is a linear pattern.
Problems
The Bellamy Water System (a fictional utility used for this set) has 3,600 lead service lines to replace. It plans to replace 300 lines per year at a cost of $6,000 per line. Under federal rules the lead action level is 15 ppb now and drops to 10 ppb in 2027, and nearly all lead lines must be replaced within 10 years. The action level is a trigger for utility action based on a system's tap samples, not a safe level; EPA's health-based goal for lead in drinking water is zero.
Write the number of lines still in the ground as a function of years, using L = 3,600 − 300t. How many lines remain after 4 years?
At 300 lines per year, in how many years would all 3,600 lines be replaced? Set 3,600 − 300t = 0 and solve for t.
The 10-year federal deadline is shorter than your answer to problem 2. What replacement rate, in lines per year, would finish all 3,600 lines in exactly 10 years?
Using the faster rate from problem 3, how many lines remain after 6 years?
At $6,000 per line, what is the total cost to replace all 3,600 lines?
At the original pace of 300 lines per year, what is the annual replacement budget in dollars?
A home's water test reads 14 ppb of lead. The action level is 15 ppb now but drops to 10 ppb in 2027. Is 14 ppb below the current level? Is it below the 2027 level? Answer each with yes or no.
By how many ppb would the 14 ppb reading exceed the 2027 action level of 10 ppb?
Compare two plans with L = 3,600 − rt: Plan A replaces 300 lines per year, Plan B replaces 360 per year. After how many years do the two plans differ by 600 remaining lines?
Explain in two or three sentences why replacing a lead service line lowers the lead that reaches a household's tap, even though the water leaves the treatment plant with no lead in it.
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: 2,400 lines. L = 3,600 − 300(4) = 3,600 − 1,200 = 2,400 lines remaining after 4 years.
Answer: 12 years. 3,600 − 300t = 0 gives 300t = 3,600, so t = 12 years.
Answer: 360 lines per year. 3,600 lines / 10 years = 360 lines per year to finish on the deadline.
Answer: 1,440 lines. L = 3,600 − 360(6) = 3,600 − 2,160 = 1,440 lines remaining after 6 years.
Answer: $21,600,000. 3,600 lines × $6,000 per line = $21,600,000 total.
Answer: $1,800,000. 300 lines per year × $6,000 per line = $1,800,000 each year.
Answer: Yes, then no. 14 ppb is below the current 15 ppb level, so yes. But 14 ppb is above the 2027 level of 10 ppb, so no, it is not below the 2027 level.
Answer: 4 ppb. 14 ppb − 10 ppb = 4 ppb above the 2027 action level.
Answer: 10 years. The gap grows by (360 − 300) = 60 lines each year, so 60t = 600 gives t = 10 years.
Answer: The treated water is lead-free when it leaves the plant, but as it sits in and passes through a lead service line, lead dissolves from the pipe into the water. Replacing that pipe with a lead-free one removes the source, so the water reaching the tap no longer picks up lead on its way in.
Differentiation
Support
Provide a two-column table with years 0 through 6 in one column so students can fill in lines remaining and see the constant drop of 300 per year before writing the function. Give the cost problems with the multiplication already set up (3,600 × 6,000) so the focus stays on interpreting the linear model.
Extension
Ask students to write the cumulative money spent as a second linear function of years at the 360-per-year pace, then find the year in which Bellamy passes the halfway point of total spending. Have them discuss why a utility might borrow money to replace lines faster than its yearly budget alone would allow.
ELL supports
Pre-teach "service line," "action level," and "replace" with a labeled diagram showing the pipe from the main to the house. Provide a sentence frame for problem 10: "The tap water picks up lead because ___, so replacing the pipe ___."
Extensions
Have students look up whether their state or water utility publishes a lead service line inventory, now required of most systems, and compare the reported number of lines to Bellamy's 3,600. A short written note can estimate, using Bellamy's per-line cost as a stand-in, what replacing the real inventory might cost and how many years it would take at a chosen pace.