Worksheet Grade 8 Engineering & Design

Belmoor's Water-Main Replacement Budget: A Linear Function Worksheet

A grade 8 worksheet where students build linear cost functions for replacing aging water mains, solve for feet within a budget, and compare two contracts.

Grade: Grade 8
Subject: Engineering & Design
Material: Worksheet
Time: 40 min
Prep: No prep
Materials: Calculator

Standards

8.F.B.4 Common Core Math

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description, a table, or a graph, and interpret them in terms of the situation modeled.

How this material aligns: Items 1 and 7 have students build a linear cost function from a setup fee and a per-foot rate, the construct-a-function work from a rate and initial value this standard names.

All materials for 8.F.B.4 →

8.EE.C.7 Common Core Math

Solve linear equations in one variable, including equations with variables on both sides, and recognizing equations with one solution, infinitely many solutions, or no solutions.

How this material aligns: Items 4 and 9 have students solve a linear equation in one variable to find the feet of main a fixed budget can replace, the equation-solving this standard requires.

All materials for 8.EE.C.7 →

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: Items 5, 6, and 10 have students judge the design against the budget and total feet of old main, defining criteria and constraints as this standard describes.

All materials for MS-ETS1-1 →

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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 write a linear cost function from a setup fee and a per-foot rate.
Students will solve a linear equation to find how many feet of main a budget can replace, and compare two contracts.

Vocabulary

Water main: A big underground pipe that carries drinking water beneath the streets to a neighborhood. Smaller service lines branch off it to reach each building.

Also: linear function, setup fee, per-foot rate, budget

Teacher overview

Many US water systems run on old pipes, and the strain shows: the ASCE 2025 report card gave the nation's drinking water a C-, and a water main breaks somewhere in the country about every two minutes. This worksheet puts students in the role of a utility planner for a fictional town that must replace aging mains on a fixed budget. Students build a linear cost function, solve for how many feet a budget covers, and compare two contractor offers. The cost model uses a setup fee that is paid once plus a steady price for each foot of pipe.

Belmoor's Water-Main Replacement Budget: A Linear Function Worksheet — Grade 8 Engineering & Design

Name: ____________________________ Date: ______________

Student Worksheet

Use these values for the problems. Suppose the town of Belmoor, a fictional town, replaces old water mains. One contractor charges a setup fee of $5,000 that is paid once, plus $60 for each foot of pipe replaced. Let f be the number of feet replaced.

Write a cost function C for replacing f feet of main with the first contractor.
Find the cost to replace 1,200 feet.
Find the cost to replace 2,500 feet.
Belmoor has a budget of $185,000 this year. How many feet of main can it replace with the first contractor? Solve for f.
Belmoor has 9,000 feet of old main in all. What percent can it replace this year with the budget? Round to the nearest whole percent.
If Belmoor replaces 3,000 feet each year, how many years will it take to replace all 9,000 feet?
A second contractor charges a lower price of $50 for each foot but a higher setup fee of $8,000. Write the cost function for the second contractor.
Using the second contractor's function, find the cost to replace 3,000 feet, and state how far under or over the $185,000 budget that is.
With the second contractor and the $185,000 budget, how many feet can Belmoor replace? Solve for f.
Using your answers, explain which contractor lets Belmoor replace more main within the same budget.

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: C = 60f + 5,000.
The per-foot rate is $60 (the slope) and the one-time setup fee is $5,000 (the starting value), so C = 60f + 5,000.
Answer: $77,000.
C = 60 x 1,200 + 5,000 = 72,000 + 5,000 = 77,000 dollars.
Answer: $155,000.
C = 60 x 2,500 + 5,000 = 150,000 + 5,000 = 155,000 dollars.
Answer: 3,000 ft.
Set C = 185,000: 60f + 5,000 = 185,000, so 60f = 180,000 and f = 3,000 feet.
Answer: 33%.
3,000 feet divided by 9,000 feet = 0.333, which rounds to about 33 percent of the town's old main.
Answer: 3 years.
9,000 feet divided by 3,000 feet each year = 3 years.
Answer: C = 50f + 8,000.
The per-foot rate is now $50 and the setup fee is $8,000, so C = 50f + 8,000.
Answer: $158,000.
C = 50 x 3,000 + 8,000 = 150,000 + 8,000 = 158,000 dollars, which is 27,000 dollars under the 185,000-dollar budget.
Answer: 3,540 ft.
Set C = 185,000: 50f + 8,000 = 185,000, so 50f = 177,000 and f = 3,540 feet.
Answer: The second contractor.
The second contractor replaces 3,540 feet for the budget, while the first replaces 3,000 feet, so the second contractor covers 540 more feet for the same 185,000 dollars.

Differentiation

Support

Give students the first cost function C = 60f + 5,000 already written and a filled example (replacing 1,000 feet) worked out, so they can focus on items 2 through 4. Allow calculators and provide a number line showing the budget as a ceiling.

Extension

Ask students to find the number of feet at which both contractors cost exactly the same, by setting 60f + 5,000 equal to 50f + 8,000 and solving, then explain which contractor is cheaper below and above that point.

ELL supports

Pre-teach setup fee as "paid one time" and per-foot rate as "paid for each foot" using a simple labeled drawing of a pipe. Provide the sentence frame "The cost is ___ for each foot plus a one-time fee of ___."