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Worksheet Grade 8 Math

Meridian Hollow Standpipe: Solving Linear Equations for Tank Drawdown

Nine problems where a fictional standpipe's fill and drain rates become linear equations, including two items on no-solution and infinite-solution cases.

Objectives

  1. Students will write a linear equation modeling a water tank's volume over time given inflow and outflow rates.
  2. Students will solve linear equations in one variable to find a specific time or volume.
  3. Students will determine whether a pair of linear equations has one solution, no solution, or infinitely many solutions.

Vocabulary

Standpipe
A standpipe is a tall, cylindrical water storage tank that sits directly on the ground instead of standing on legs like a water tower. It stores water and helps maintain steady pressure in the pipes nearby.

Also: net rate, linear equation, no solution, infinitely many solutions

Teacher overview

Meridian Hollow (a fictional town used for this worksheet) draws its water pressure from a single standpipe, a tall, narrow elevated tank. This worksheet turns the tank's fill and drain rates into linear equations, the same modeling move an operator makes when planning a maintenance shutdown. A common misconception in this kind of problem is treating the inflow and outflow rates as if they simply add together instead of working in opposite directions; students need to subtract the outflow rate from the inflow rate to find the tank's true net rate before writing an equation.

Meridian Hollow Standpipe: Solving Linear Equations for Tank Drawdown — Grade 8 Math
Name: ____________________________ Date: ______________

Student Worksheet

Meridian Hollow's standpipe supplies the west side of town. Two pumps can add a combined 400 gallons of water per minute, and during a typical weekday the town draws water out of the tank at a steady rate. Use the given rates for each problem below.

  1. At 6:00 a.m., the standpipe holds 180,000 gallons. The pumps run and add 400 gallons per minute while the town draws water out at 250 gallons per minute. Write an equation for the tank's volume, V, in gallons, after t minutes.
  2. Using your equation from problem 1, how many gallons are in the tank after 45 minutes?
  3. Using your equation from problem 1, solve for t: after how many minutes will the tank reach 213,000 gallons?
  4. At 1:00 p.m., crews shut off both pumps for scheduled maintenance. The tank holds 180,000 gallons at the moment they shut off, and water continues draining out at 250 gallons per minute with no pump adding water. Write an equation for the tank's volume, V, after t minutes during this maintenance window, and use it to find the volume after 20 minutes.
  5. Using your equation from problem 4, solve for t: after how many minutes will the tank's volume drop to the minimum operating level of 45,000 gallons?
  6. A second standpipe, Tank B, also begins maintenance at 1:00 p.m. with a starting volume of 140,000 gallons, draining at the same rate of 250 gallons per minute with no inflow. Write Tank B's equation for its volume after t minutes. Then set Tank A's equation (problem 4) equal to Tank B's equation and solve. How many solutions does the resulting equation have, and what does that tell you about whether the two tanks are ever at the same volume at the same time?
  7. A third standpipe, Tank C, begins maintenance at exactly 180,000 gallons and drains at 250 gallons per minute too, matching Tank A in problem 4 in every detail. Set Tank A's equation equal to Tank C's equation and solve. How many solutions does this equation have this time, and why does that make sense given how Tank C was described?
  8. That evening, the pumps come back online at 400 gallons per minute while the town's demand drops to 150 gallons per minute overnight. Starting from a volume of 90,000 gallons, find the tank's volume after 2 hours (120 minutes) of this overnight rate.
  9. Using the overnight rate from problem 8, solve for t: how many minutes will it take the tank to refill to its full capacity of 240,000 gallons?

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: V = 180,000 + 150t.
    The net rate is inflow minus outflow: 400 − 250 = 150 gallons per minute added overall, starting from 180,000 gallons.
  2. Answer: 186,750 gal.
    V = 180,000 + 150(45) = 180,000 + 6,750 = 186,750 gallons.
  3. Answer: 220 minutes.
    180,000 + 150t = 213,000 → 150t = 33,000 → t = 220.
  4. Answer: 175,000 gal.
    With no inflow, V = 180,000 − 250t. At t = 20: V = 180,000 − 5,000 = 175,000 gallons.
  5. Answer: 540 minutes.
    180,000 − 250t = 45,000 → −250t = −135,000 → t = 540 minutes, which is 9 hours.
  6. Answer: no solution.
    Tank B's equation is V = 140,000 − 250t. Setting 180,000 − 250t = 140,000 − 250t and adding 250t to both sides gives 180,000 = 140,000, a false statement. There is no solution, which means the two tanks are never at the same volume at the same time; both drain at the same rate, so the 40,000-gallon gap between them never closes.
  7. Answer: infinitely many solutions.
    Tank C's equation is identical to Tank A's: V = 180,000 − 250t. Setting the two equal gives 180,000 − 250t = 180,000 − 250t, which simplifies to 0 = 0, true for every value of t. This makes sense because Tank C was described with the exact same starting volume and drain rate as Tank A, so the two are really the same equation and are equal at every single minute, not just once.
  8. Answer: 120,000 gal.
    The overnight net rate is 400 − 150 = 250 gallons per minute added. V = 90,000 + 250t. At t = 120: V = 90,000 + 30,000 = 120,000 gallons.
  9. Answer: 600 minutes.
    90,000 + 250t = 240,000 → 250t = 150,000 → t = 600 minutes, which is 10 hours.

Differentiation

Support (below grade level)

Give students a labeled arrow diagram showing "inflow − outflow = net rate" before problem 1, and provide the net rate already calculated (150 gallons per minute) for problems 1 through 3 so students can focus on writing and solving the equation itself. Allow a calculator for all arithmetic.

Extension (above grade level)

Ask students to describe in words a piecewise model for the three rate periods in problems 1, 4, and 8, each starting from the volume given in that problem, and to explain why one linear equation cannot model all three periods at once.

ELL supports

Pre-teach "inflow," "outflow," "net rate," and "standpipe" with a simple labeled diagram of a tank with an arrow pointing in (pump) and an arrow pointing out (town demand). Provide a sentence frame for problems 6 and 7: "This equation has ___ solution(s), which means ___."

Extensions

Have students research (using only the rates given in this worksheet, not outside sources) what net rate a third pump would need to add for Meridian Hollow's standpipe to refill from 90,000 gallons to full capacity (240,000 gallons) in exactly 4 hours instead of 10, and have them write and solve the equation that finds it.