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Worksheet Grade 8 Engineering & Design

Foxhollow School Rainwater Cistern: A Linear Function Worksheet

A grade 8 worksheet modeling a rainwater cistern's fill, hold, and drain phases as linear functions of time, interpreting the rate of change from each phase.

Objectives

  1. Students will build a table of values and a function rule for a tank's fill phase, identifying rate of change and initial value.
  2. Students will write a function rule for a tank's drain phase and use it to find when the tank empties.
  3. Students will sketch a graph showing a tank's volume through its fill, hold, and drain phases.

Vocabulary

Cistern
A large tank that stores rainwater collected from a roof so it can be used later, usually for watering plants.
Green infrastructure
Green infrastructure uses soil and plants, like rain gardens, green roofs, and street trees, to soak up rainwater where it falls instead of piping it away. It copies the way nature handles rain.

Also: function, rate of change, initial value

Teacher overview

The fictional Foxhollow Middle School installed an underground rainwater harvesting cistern as part of a green-infrastructure retrofit, so the school garden's irrigation system no longer depends only on the town's treated water supply. This worksheet has students model the cistern's volume during a storm, tracking three phases: the roof runoff filling the cistern, a short waiting period before the irrigation controller turns on, and the slow drain that follows, and it treats each phase as its own linear function of elapsed time.

Foxhollow School Rainwater Cistern: A Linear Function Worksheet — Grade 8 Engineering & Design
Name: ____________________________ Date: ______________

Student Worksheet

Foxhollow's fictional cistern starts empty just before a storm. During the storm, roof runoff fills the cistern at a constant rate of 150 gallons per minute (gpm) for 20 minutes, when it reaches its full capacity. The cistern then holds steady for 20 minutes while the irrigation controller waits to activate. After that, the controller drains the cistern into the garden's drip lines at a constant rate of 5 gpm.

  1. Complete this table for the fill phase (0 to 20 minutes), using the fill rate of 150 gpm.

    Time, t (minutes) Volume, V (gallons)
    0
    5
    10
    15
    20

  2. Explain why the cistern's volume during the fill phase is a function of elapsed time. In your explanation, state what it would mean if a single moment in time had two different possible volumes.

  3. Write a function rule V(t) for the fill phase, where 0 ≤ t ≤ 20. State the rate of change and the initial value in this equation, and explain what each one means in the context of the cistern filling.

  4. During the hold phase (20 to 40 minutes), what is the cistern's volume? Explain why the rate of change is 0 gallons per minute during this phase.

  5. Write a function rule V(t) for the drain phase, valid for t ≥ 40 minutes, using the cistern's full capacity and the 5 gpm drain rate. Use your function to find V(70), the volume at 70 minutes.

  6. Sketch a graph of the cistern's volume from t = 0 minutes until the cistern is completely empty. Describe the three sections of your graph in words: what the line does during the fill phase, the hold phase, and the drain phase.

  7. Using your drain-phase function from item 5, find the exact time t, in minutes, when the cistern becomes completely empty.

  8. Foxhollow's garden club wants the cistern to keep the irrigation system running through a 3-day dry spell, which is 4,320 minutes of drain time. Using your answer from item 7, does the current cistern and drain rate meet this goal? Explain using your calculation.

  9. Challenge. What drain rate, in gallons per minute, would let the cistern's full 3,000-gallon capacity last exactly 4,320 minutes? Round to the nearest hundredth of a gallon per minute, and explain in one sentence why a rate that slow might not be practical for watering a garden.

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: 0, 750, 1,500, 2,250, 3,000 gallons.
    Each value equals 150 gpm × the number of minutes: 150(0) = 0; 150(5) = 750; 150(10) = 1,500; 150(15) = 2,250; 150(20) = 3,000.

  2. Answer: The volume is a function of time because every single moment during the fill phase has exactly one corresponding volume.
    If a moment in time had two different possible volumes, the relationship would not be a function; a real cistern cannot be at two different fill levels at the same instant.

  3. Answer: V(t) = 150t.
    The rate of change is 150 gallons per minute, the speed at which the cistern fills. The initial value is 0 gallons, the cistern's volume at t = 0, before the storm begins.

  4. Answer: 3,000 gallons; the rate of change is 0 because the volume is not changing.
    During the hold phase the cistern is neither filling nor draining, so its volume stays constant at the full 3,000-gallon capacity reached at the end of the fill phase, and a constant volume means a rate of change of 0 gallons per minute.

  5. Answer: V(t) = 3,000 − 5(t − 40); V(70) = 2,850 gallons.
    The cistern starts the drain phase at its full 3,000 gallons and loses 5 gallons for every minute past t = 40. At t = 70: V(70) = 3,000 − 5(70 − 40) = 3,000 − 5(30) = 3,000 − 150 = 2,850 gallons.

  6. Answer: A correctly sketched graph rising from (0, 0) to (20, 3,000), staying flat from (20, 3,000) to (40, 3,000), then falling from (40, 3,000) toward the horizontal axis.
    The fill phase should show a straight line rising steeply from the origin; the hold phase should show a flat horizontal line at 3,000 gallons; the drain phase should show a straight line falling slowly, since the drain rate (5 gpm) is much slower than the fill rate (150 gpm).

  7. Answer: t = 640 minutes.
    Set V(t) = 0: 3,000 − 5(t − 40) = 0. Add 5(t − 40) to both sides: 3,000 = 5(t − 40). Divide both sides by 5: 600 = t − 40. Add 40 to both sides: t = 640.

  8. Answer: No, it does not meet the goal.
    The cistern's drain phase begins at t = 40 and ends at t = 640, so the drain phase itself lasts only 640 − 40 = 600 minutes, far short of the 4,320 minutes (3 days) the garden club wants.

  9. Answer: About 0.69 gallons per minute.
    3,000 gallons ÷ 4,320 minutes = 0.694..., which rounds to 0.69 gpm. A drain rate that slow would deliver only a trickle of water to the drip lines, which may not be enough to keep plants adequately watered during a hot, dry stretch.

Differentiation

Support

Provide the fill-phase table in item 1 already completed, and give students the drain-phase function from item 5 already written so they only need to substitute t = 70 to find V(70). Allow a calculator for all computations.

Extension

Ask students to find a fill rate, instead of the given 150 gpm, that would let the cistern reach its 3,000-gallon capacity in exactly 15 minutes rather than 20, and have them write the new fill-phase function.

ELL supports

Provide a simple diagram of the cistern with an arrow labeled "fill" pointing in during the storm and an arrow labeled "drain" pointing out toward the garden, alongside a blank version of the three-part graph shape (rising, flat, falling) for students to trace and label. Allow item 2, 4, and 8's explanations to be written as short phrases with a provided sentence starter: "The rate of change is 0 because ___."

Every figure in this scenario is fictional, chosen so students work with realistic engineering numbers.

Extensions

Have students design an alternative drain schedule for the cistern, choosing their own constant drain rate, and calculate how many minutes their new rate would let the cistern's full capacity last, comparing their result to the 600 minutes the original 5 gpm rate provides.