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Math problem set Grade 8 Math

Cobalt Springs Pump Station: A System of Equations for Two Tanks

A grade 8 math problem set where students write linear equations for two storage tanks and solve the system to find when their volumes are equal.

Objectives

  1. Students will write a linear equation modeling a storage tank's volume over time from its starting volume and fill rate.
  2. Students will solve a system of two linear equations to find when two tanks hold equal volumes of water.
  3. Students will interpret the slope and starting value of a tank equation in terms of pump rate and initial water level.

Vocabulary

Key words: system of equations, rate of change, initial value

Teacher overview

Cobalt Springs (a fictional town used for this problem set) is filling two storage tanks at once: Well A feeds Tank 1, and Well B feeds Tank 2. Tank 1 starts with less water but fills faster, while Tank 2 starts with more water but fills more slowly, so at some point the two tanks hold exactly the same volume. This problem set has students write a linear equation for each tank and solve the resulting system to find that moment. A common misconception is assuming the tank that starts with more water always stays ahead; the equations show that a faster constant rate can overtake a head start, which is exactly what solving the system reveals.

Cobalt Springs Pump Station: A System of Equations for Two Tanks — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

Cobalt Springs' public works department (fictional, created for this problem set) is monitoring two tanks fed by two different wells. Use the information below for every problem, where x stands for minutes elapsed and y stands for gallons in the tank.

  • Tank 1 starts with 400 gallons already inside it. Well A's pump adds 25 gallons every minute.
  • Tank 2 starts with 700 gallons already inside it. Well B's pump adds 10 gallons every minute.
  1. Write an equation for Tank 1's volume y, in gallons, after x minutes.
  2. Write an equation for Tank 2's volume y, in gallons, after x minutes.
  3. Use your two equations to find the number of minutes, x, at which both tanks contain the same volume of water. Show your work solving the system.
  4. What is that shared volume, in gallons, when the two tanks are equal? Use both equations to check your answer.
  5. At x = 10 minutes, which tank holds more water? Use your equations to justify your answer.
  6. At x = 30 minutes, which tank holds more water? Explain why the tank with the faster pump eventually pulls ahead even though it started with less water.
  7. Cobalt Springs' town engineer wants Tank 1 to reach 1,000 gallons. Using Tank 1's equation, how many minutes will that take?
  8. Cobalt Springs wants Tank 1 to catch up to Tank 2's volume in only 10 minutes instead of 20, by adding a booster pump alongside Well A. Using Tank 2's volume at x = 10 minutes as the target, what total fill rate, in gallons per minute, would Tank 1 need? How much faster is that than Well A's original rate of 25 gallons every minute?

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 = 400 + 25x.
    Tank 1 starts at 400 gallons (the initial value) and gains 25 gallons every minute (the rate of change), so y = 400 + 25x.
  2. Answer: y = 700 + 10x.
    Tank 2 starts at 700 gallons and gains 10 gallons every minute, so y = 700 + 10x.
  3. Answer: 20 minutes.
    Set the two expressions equal: 400 + 25x = 700 + 10x. Subtract 10x from both sides: 400 + 15x = 700. Subtract 400 from both sides: 15x = 300. Divide by 15: x = 20 minutes.
  4. Answer: 900 gallons.
    Substitute x = 20 into either equation. Tank 1: 400 + 25(20) = 400 + 500 = 900 gallons. Tank 2: 700 + 10(20) = 700 + 200 = 900 gallons. Both equations agree.
  5. Answer: Tank 2 holds more water at x = 10 minutes.
    Tank 1: 400 + 25(10) = 650 gallons. Tank 2: 700 + 10(10) = 800 gallons. 800 is greater than 650, so Tank 2 is ahead before the tanks become equal.
  6. Answer: Tank 1 holds more water at x = 30 minutes.
    Tank 1: 400 + 25(30) = 400 + 750 = 1,150 gallons. Tank 2: 700 + 10(30) = 700 + 300 = 1,000 gallons. Tank 1's faster rate of 25 gallons a minute, compared with Tank 2's 10 gallons a minute, lets it overtake Tank 2's head start once enough time passes.
  7. Answer: 24 minutes.
    Set 1,000 = 400 + 25x. Subtract 400 from both sides: 600 = 25x. Divide by 25: x = 24 minutes.
  8. Answer: 40 gallons per minute.
    Tank 2's volume at x = 10 minutes is 700 + 10(10) = 800 gallons, the new target for Tank 1. Setting 400 + 10r = 800 gives 10r = 400, so r = 40 gallons every minute, which is 40 − 25 = 15 gallons every minute faster than Well A's original rate.

Differentiation

Support (below grade level)

Give students a table of Tank 1 and Tank 2's volumes already calculated at x = 0, 10, 20, and 30 minutes so they can see the pattern before solving problem 3 algebraically, and allow them to solve the system by testing values in the table instead of by elimination.

Extension (above grade level)

Ask students to graph both equations on the same coordinate grid, by hand or with graphing software, and confirm that the two lines cross at the point they calculated in problems 3 and 4.

ELL supports

Pre-teach "rate," "initial value," and "system of equations" using a simple two-runner race analogy, one runner with a head start and one who runs faster, before introducing the tank scenario. Allow students to explain problem 6's reasoning with a labeled sketch of the two lines crossing instead of a full paragraph.

Extensions

Have students choose new starting volumes and rates for a third tank, write its equation, and determine algebraically whether and when it would ever hold the same volume as Tank 1.