Math problem set Grade 8 Engineering & Design

Sunmarch Ridge's Solar Well Pump: Building a Linear Function

A grade 8 set where students build and use linear functions for a solar well pump, combining a steady fill rate, a starting amount, and daily household use.

Grade: Grade 8
Subject: Engineering & Design
Material: Math problem set
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: Problems 1 through 6 have students build the linear function for tank gallons from a starting amount and fill rate, then find the net rate of change once household use is subtracted and rebuild the function, the construct-a-function and rate-of-change work this standard names.

All materials for 8.F.B.4 →

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: Problem 4 and problem 8 have students test the design against tank capacity and cloudy-day limits, defining the criteria and constraints 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 function in the form y = mx + b for water pumped into a tank over time.
Students will find the net rate of change when pump inflow and household use happen at the same time.

Vocabulary

Key words: linear function, rate of change, net rate, initial value

Teacher overview

Sunmarch Ridge is a fictional hillside community that runs its well pump on solar power. On a sunny day the pump lifts water into a storage tank at a steady rate, and the tank already holds some water when the sun comes up. This set has students build a linear function for the gallons in the tank, then adjust it once household use is subtracted. A common misconception is mixing up the rate of change with the starting amount; the gallons added each hour is the slope, and the gallons already in the tank at sunrise is the y-intercept, so students should label which number is which before writing the function.

Sunmarch Ridge's Solar Well Pump: Building a Linear Function — Grade 8 Engineering & Design

Name: ____________________________ Date: ______________

Problems

Use these design values for every problem. Suppose the Sunmarch Ridge tank, a fictional tank, holds 400 gallons at sunrise before the pump starts. When the sun is strong, the solar pump adds 160 gallons of water each hour. Let h be the number of hours of strong sun.

How many gallons are in the tank at sunrise, before the pump starts?
Write a linear function for the gallons G in the tank after h hours of strong sun.
Use your function to find the gallons in the tank after 5 hours of sun.
The tank is full at 2,000 gallons. After how many hours of strong sun does the tank fill?
During the day the community's homes draw 90 gallons of water each hour. What is the net rate of change, meaning gallons added minus gallons used, each hour?
Write the net linear function for the gallons in the tank after h hours, using the net rate from problem 5.
Using the net function, how many gallons are in the tank after 8 hours of sun?
Challenge. On a cloudy day the pump adds only 100 gallons each hour while the homes still use 90 gallons each hour. After how many hours does the tank hold 60 more gallons than it did at sunrise?

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: 400 gal.
The problem states the tank already holds 400 gallons at sunrise, which is the starting amount before any pumping.
Answer: G = 160h + 400.
The pump adds 160 gallons each hour, so 160 is the rate of change (slope). The 400 gallons at sunrise is the initial value (y-intercept). The function is G = 160h + 400.
Answer: 1,200 gal.
G = 160 x 5 + 400 = 800 + 400 = 1,200 gallons after 5 hours.
Answer: 10 hours.
Set G = 2,000: 160h + 400 = 2,000, so 160h = 1,600 and h = 10 hours of strong sun.
Answer: 70 gal per hour.
The pump adds 160 gallons and the homes use 90 gallons each hour, so the net rate is 160 minus 90 = 70 gallons per hour.
Answer: G = 70h + 400.
The net rate of change is 70 gallons per hour and the starting amount is still 400 gallons, so the net function is G = 70h + 400.
Answer: 960 gal.
G = 70 x 8 + 400 = 560 + 400 = 960 gallons after 8 hours.
Answer: 6 hours.
On the cloudy day the net rate is 100 minus 90 = 10 gallons per hour. To gain 60 gallons above sunrise, solve 10h = 60, so h = 6 hours.

Differentiation

Support

Give students a filled-in example function, such as G = 50h + 100, and have them identify the slope and y-intercept before writing their own in problem 2. Provide a two-column table of h and G for problem 3 so they can check the pattern of adding 160 each hour.

Extension

Ask students to graph both the sunny-day net function and the cloudy-day net function on the same axes, then explain what the different slopes show about how fast the tank fills on each kind of day.

ELL supports

Pre-teach slope as "how much is added each hour" and y-intercept as "how much is there at the start" using a simple sketch of the tank. Provide the sentence frame "The tank starts with ___ gallons and gains ___ gallons each hour" to help students describe their function in words.