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Math problem set Grade 8 Careers & CTE

Winnowbrook's Lead Service Lines: Crew Cost Versus Contractor Bid

A grade 8 problem set where students write and solve a system of equations to compare an in-house replacement crew with a contractor bid for lead service lines.

Objectives

  1. Students will write linear cost equations for two ways of staffing a service line replacement program.
  2. Students will solve a system of two linear equations and interpret the solution as a break-even number of service lines.
  3. Students will explain why the cheaper option changes depending on how many lines a utility replaces.

Vocabulary

Service line
A service line is the small pipe that connects the water main under the street to the plumbing inside one home or building. It is the last leg of the trip water makes to your tap.

Also: system of equations, break-even, fixed cost, rate of change

Teacher overview

EPA estimates that about 4 million lead service lines remain in US water systems, and EPA's Lead and Copper Rule Improvements require utilities to replace nearly all of them within ten years. That deadline has turned service line replacement into steady work for distribution crews, and it has forced small utilities to decide whether to build their own crew or hire a contractor line by line. This problem set puts students in the seat of a project coordinator for the fictional town of Winnowbrook, who has to answer that question with a pair of cost equations. The misconception this set is built to correct is the assumption that the option with the lower price per line is automatically the cheaper option; students who ignore the fixed yearly cost of owning equipment and paying a crew will pick the wrong plan for a large replacement program.

Winnowbrook's Lead Service Lines: Crew Cost Versus Contractor Bid — Grade 8 Careers & CTE
Name: ____________________________ Date: ______________

Problems

Suppose Winnowbrook Water Works, a fictional utility, must replace the lead service lines on its older streets. The coordinator has two staffing plans, and all figures below are invented for this problem set. Let n stand for the number of service lines replaced in one year.

  • Plan A, in-house crew. The utility pays $180,000 per year for the excavator, the truck, and the crew's wages whether they replace one line or one hundred, plus $4,200 in pipe, fittings, and restoration for each line replaced.
  • Plan B, contractor. The utility pays nothing up front and is billed $6,600 for each line the contractor replaces.
  1. Write an equation for the total yearly cost y of Plan A in terms of n. Name the fixed cost and the rate of change.
  2. Write an equation for the total yearly cost y of Plan B in terms of n.
  3. What does Plan A cost if the crew replaces 40 lines in a year? Show your substitution.
  4. What does Plan B cost for those same 40 lines? Show your substitution.
  5. At 40 lines, how much more does Plan A cost than Plan B?
  6. Solve the system of your two equations to find the number of lines at which both plans cost the same. Show every step.
  7. What is the total cost of either plan at that break-even number of lines?
  8. Winnowbrook's engineer says the town can replace 120 lines next year. What would Plan A cost for 120 lines?
  9. What would Plan B cost for those same 120 lines?
  10. Using the rate of change in each equation, explain why Plan B is cheaper for small numbers of lines while Plan A is cheaper for large numbers.

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

Answer Key
  1. Answer: y = 180,000 + 4,200n.
    The fixed cost is $180,000 per year, which the utility pays no matter how many lines are replaced. The rate of change is $4,200 per line.
  2. Answer: y = 6,600n.
    Plan B has no fixed cost, so the equation is a single term. The rate of change is $6,600 per line.
  3. Answer: $348,000. y = 180,000 + 4,200(40) = 180,000 + 168,000 = 348,000.
  4. Answer: $264,000. y = 6,600(40) = 264,000.
  5. Answer: $84,000.
    348,000 - 264,000 = 84,000, so Plan A costs $84,000 more at 40 lines.
  6. Answer: 75 lines.
    Set the two expressions equal: 180,000 + 4,200n = 6,600n. Subtract 4,200n from both sides: 180,000 = 2,400n. Divide both sides by 2,400: n = 75.
  7. Answer: $495,000.
    Plan A: 180,000 + 4,200(75) = 180,000 + 315,000 = 495,000. Plan B: 6,600(75) = 495,000. The two agree, which confirms the solution to problem 6.
  8. Answer: $684,000. y = 180,000 + 4,200(120) = 180,000 + 504,000 = 684,000.
  9. Answer: $792,000. y = 6,600(120) = 792,000.
  10. Answer: Plan A adds only $4,200 per line while Plan B adds $6,600 per line.
    Plan A starts $180,000 behind because of its fixed cost, so for small values of n that head start for Plan B still decides the comparison. Each additional line narrows the gap by 6,600 - 4,200 = 2,400 dollars. After 75 lines the accumulated savings of $2,400 per line have erased the $180,000 fixed cost, and beyond that point Plan A stays cheaper and the gap keeps widening.

Differentiation

Support

Give students the two equations for this fictional scenario already written, and have them begin at problem 3 so the arithmetic of substitution comes before the algebra of solving. A filled-in table with columns for 20, 40, 60, 80, and 100 lines lets students see the crossing point numerically before they solve for it.

Extension

Ask students to find the contractor price per line that would move the break-even point of this invented scenario down to 50 lines, then verify their price by substituting it back into both equations.

ELL supports

Pre-teach "service line," "fixed cost," and "break-even" with a labeled sketch showing the pipe that runs from the water main to a house. Allow problem 10 to be answered with the two rates of change plus a short phrase rather than a full paragraph.

Extensions

Have students graph both plans of this fictional comparison on the same axes with lines replaced on the horizontal axis and dollars on the vertical axis, then mark the intersection point and label the region where each plan is cheaper. Students who finish early can research what their own state requires utilities to report about lead service line inventories, and describe which job title would prepare that report.