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Math problem set Grade 9 Math

Larkspur Flats Well Drawdown: A Grade 9 Linear Equations Problem Set

Eight algebra problems that model a subdivision's groundwater well as a linear equation, using drawdown rate, a safety shutoff, and unit conversions.

Objectives

  1. Write a linear equation modeling a well's pumping water level as a function of pumping time.
  2. Solve linear equations and inequalities to find when a well reaches a stated safety shutoff level.
  3. Convert units, including hours to hours-and-minutes and gallons per minute to million gallons per day.

Vocabulary

Drawdown
Drawdown is how far the water level in a well drops when a pump pulls water out. The harder and longer you pump, the bigger the drawdown gets.
Cone of depression
A cone of depression is the dip in the water table that forms in a circle around a well while it is pumping, like a shallow bowl scooped out of the underground water surface. The harder a well pumps, the wider and deeper this dip becomes.
Aquifer
An aquifer is an underground layer of rock, sand, or gravel that holds water in the tiny spaces between its grains. Wells reach down into aquifers to bring water up to the surface.

Teacher overview

Larkspur Flats (a fictional subdivision invented for this problem set) depends on two groundwater wells to supply a growing neighborhood. When a well pump runs, it draws the water level inside the well down below its resting depth, a process called drawdown. This set uses a simplified constant-rate model of drawdown. Real aquifers often draw down in a more complex, curving pattern, but a constant rate is a reasonable approximation for a first algebra model. A common misconception in this math is assuming that a smaller "feet below ground" number means the water has dropped further, when the opposite is true: a larger number means the water sits deeper, farther from the surface, and during recovery that same number gets smaller as the water level rises back up.

Larkspur Flats Well Drawdown: A Grade 9 Linear Equations Problem Set — Grade 9 Math
Name: ____________________________ Date: ______________

Problems

Larkspur Flats (the fictional subdivision used throughout this set) is served by Well 3 and Well 4. Before pumping starts, Well 3's static water level sits 60 feet below ground. When Well 3 pumps continuously, its water level drops at a constant rate of 3.5 feet per hour. To protect the pump, the well shuts off automatically once its pumping water level reaches 150 feet below ground.

  1. Write a linear equation for Well 3's pumping water level, L, in feet below ground, after t hours of continuous pumping.
  2. Using your equation from problem 1, find how many hours of continuous pumping it takes for Well 3 to reach its 150-foot safety shutoff. Round to the nearest tenth of an hour.
  3. Convert your unrounded answer from problem 2 (before rounding to the tenth) into hours and minutes, rounding the minutes to the nearest whole minute.
  4. Well 4 has a static water level of 45 feet and a constant drawdown rate of 2 feet per hour, with the same 150-foot safety shutoff. Write Well 4's linear equation and find how many hours of continuous pumping it takes to reach the shutoff level.
  5. Compare Well 3 and Well 4: how many more hours can Well 4 run continuously before reaching its shutoff level than Well 3? Round to the nearest tenth of an hour.
  6. Well 3's pump moves water at 250 gallons per minute (250 gpm). Express this flow rate in gallons per day, then convert it to million gallons per day (MGD).
  7. Write an inequality representing the number of hours, t, that Well 3 can pump continuously and remain within its safe operating range, that is, before its pumping water level reaches the 150-foot shutoff. State the viable range of t.
  8. Challenge. After Well 3's pump shuts off automatically at 150 feet, the aquifer recharges, and the water level rises back toward the static level, at a constant rate of 0.5 feet per hour. Write an equation for the water level during recovery, using the shutoff moment as your starting point, and find how many hours of recovery it takes to return to a level of 100 feet below ground, the point at which the utility is willing to resume pumping.

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: L(t) = 60 + 3.5t.
    Drawdown equals the pumping water level minus the static water level, so the pumping water level equals the static level plus the drawdown: 60 feet of static level plus 3.5 feet per hour of drawdown, times t hours.
  2. Answer: 25.7 hours.
    Set L(t) = 150: 150 = 60 + 3.5t, so 90 = 3.5t, and t = 90 ÷ 3.5 = 25.714..., which rounds to 25.7 hours.
  3. Answer: 25 hours 43 minutes (1,543 minutes total).
    25.714... hours is 25 whole hours plus 0.714... of an hour. 0.714... × 60 minutes = 42.86..., which rounds to 43 minutes, giving 25 hours 43 minutes, or 25 × 60 + 43 = 1,543 minutes total.
  4. Answer: 52.5 hours (L(t) = 45 + 2t).
    Set 45 + 2t = 150: 105 = 2t, so t = 52.5 hours exactly.
  5. Answer: 26.8 hours.
    52.5 − 25.714... = 26.785..., which rounds to 26.8 hours. Well 4 can run about 26.8 hours longer than Well 3 before reaching its shutoff level.
  6. Answer: 0.36 MGD.
    250 gallons/minute × 1,440 minutes/day = 360,000 gallons/day. Since one million gallons per day (MGD) equals 1,000,000 gallons/day, 360,000 ÷ 1,000,000 = 0.36 MGD.
  7. Answer: 25.7 hours (upper bound; viable range 0 ≤ t ≤ 25.7).
    The inequality is 60 + 3.5t ≤ 150. Subtracting 60 from both sides gives 3.5t ≤ 90, and dividing by 3.5 gives t ≤ 25.7 (rounded). Combined with t ≥ 0, since pumping time cannot be negative, the viable range is 0 to about 25.7 hours.
  8. Answer: 100 hours (R(t) = 150 − 0.5t).
    At the moment of shutoff, recovery time t = 0 and the level is 150 feet. Recovery decreases that number by 0.5 feet per hour: R(t) = 150 − 0.5t. Setting R(t) = 100 gives 150 − 0.5t = 100, so 0.5t = 50, and t = 100 hours.

Differentiation

Support (below grade level)

Give students the completed equation for Well 3 (problem 1) already written, so they can focus on solving rather than building the equation, and provide a labeled number line showing "smaller number = closer to the surface, larger number = deeper" before problem 1. Allow a calculator for all division steps.

Extension (above grade level)

Ask students to write a piecewise equation that models Well 3's full pump-and-recover cycle if the utility restarts pumping the moment the well recovers to 100 feet, then predict whether Well 3 could complete a second full pump-and-recover cycle within a 5-day period, showing the reasoning that supports the prediction.

ELL supports

Pre-teach drawdown, static level, and aquifer with a labeled cross-section diagram of a well showing the ground surface, the static water level, and the pumping water level marked as three distinct horizontal lines. Allow students to show their work for problems 2, 4, and 8 using the diagram plus numbers rather than full written sentences.

Extensions

Contact a local water utility or well-drilling company to ask whether they can share a real, anonymized drawdown reading from a nearby well. Compare the reported drawdown rate to Well 3's and Well 4's rates in this set, and discuss what factors (aquifer type, well depth, pumping rate) might explain any difference.