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Math problem set Grade 9 Math

Crestline Reservoir's Drought Drawdown: Equations and Rationing Limits

A grade 9 problem set where students write and solve one-variable equations for a drought-stricken reservoir's drawdown and a household's rationing constraints.

Objectives

  1. Students will write and solve a one-variable equation modeling a reservoir's declining volume during a drought.
  2. Students will represent a set of usage constraints as a system of inequalities and determine whether a given combination is viable.

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.

Also: critical minimum, constraint, viable solution

Teacher overview

The fictional Crestline Reservoir is losing water during a drought, and its managers need to know exactly how many days remain before the reservoir drops to a critical minimum that would trigger emergency rationing. This problem set has students write a one-variable equation for that drawdown, solve it for the number of days, and then see how a mid-drought conservation measure changes the timeline. The second half of the set shifts scale, from the whole reservoir down to a single household, where the rationing rule becomes a system of inequalities a family's daily water use must satisfy. A common misconception in this kind of problem is treating "total use under budget" as the only requirement, when a second constraint, such as a minimum for basic health needs, can still be violated even when the total looks fine.

Crestline Reservoir's Drought Drawdown: Equations and Rationing Limits — Grade 9 Math
Name: ____________________________ Date: ______________

Problems

The fictional Crestline Reservoir currently holds 480 million gallons and is declining at a constant rate of 6 million gallons per day during the drought.

  1. Write an equation for the reservoir's volume, V (in million gallons), after d days of decline at this constant rate.

  2. Using your equation from problem 1, solve for the number of days until the reservoir reaches its critical minimum of 300 million gallons.

  3. Water managers want to trigger emergency rationing 10 days before the reservoir actually reaches its critical minimum, to give the town time to adjust. On which day (counting from today, day 0) should rationing begin?

  4. Conservation measures begin on the day rationing starts, your answer to problem 3. These measures cut the decline rate in half, from 6 million gallons per day to 3 million gallons per day. First, find the reservoir's volume on that day using your equation from problem 1. Then write a new equation for the volume. Use x for the number of days since rationing started, so x = 0 is the day rationing began.

  5. Using your new equation from problem 4, solve for how many additional days (x) it will take the reservoir to reach the 300 million gallon critical minimum under the reduced decline rate.

  6. Add your answer from problem 3 to your answer from problem 5. This gives the total number of days until the reservoir would reach its critical minimum with conservation measures in place. Compare this total to your answer from problem 2, which assumed no conservation measures. How many extra days does the conservation measure buy the town?

  7. During the rationing period, each Crestline household tracks two values: indoor use, x, and outdoor use, y, both in gallons per day. The combined total must not exceed 150 gallons per day. Indoor use must also be at least 70 gallons per day to meet basic health needs. Write this rationing rule as a system of inequalities in x and y.

  8. One household uses 90 gallons per day indoors and 50 gallons per day outdoors. Check this combination against both inequalities from problem 7. Is it a viable combination under the rationing rule?

  9. A second household uses 60 gallons per day indoors and 80 gallons per day outdoors. Check this combination against both inequalities from problem 7. Is it viable? If not, explain specifically which constraint it fails, even though its total use is under the 150-gallon limit.

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 = 480 − 6d.
    The reservoir starts at 480 million gallons and loses 6 million gallons for each day, d, that passes, giving V = 480 − 6d.

  2. Answer: 30 days.
    Setting V = 300: 480 − 6d = 300. Subtracting 480 from both sides: -6d = -180. Dividing by -6: d = 30 days.

  3. Answer: day 20.
    Rationing should begin 10 days before day 30, the day the reservoir would reach its critical minimum: 30 − 10 = day 20.

  4. Answer: 360 million gallons on day 20; new equation V = 360 − 3x.
    Substituting d = 20 into V = 480 − 6d: V = 480 − 6(20) = 480 − 120 = 360 million gallons. With the reduced rate of 3 million gallons per day starting from this point, the new equation is V = 360 − 3x.

  5. Answer: 20 days.
    Setting V = 300 in the new equation: 360 − 3x = 300. Subtracting 360 from both sides: -3x = -60. Dividing by -3: x = 20 days.

  6. Answer: 40 days total; 10 extra days.
    Adding day 20 (when rationing began) to the 20 additional days from problem 5 gives day 40 as the total time until the reservoir reaches its critical minimum with conservation measures. Without conservation, problem 2 found the reservoir would reach that point on day 30. The conservation measure buys the town 40 − 30 = 10 extra days.

  7. Answer: x + y ≤ 150 and x ≥ 70.
    The combined-use limit gives x + y ≤ 150, and the indoor health minimum gives x ≥ 70. (Outdoor use is also assumed non-negative, y ≥ 0, though this is not stated explicitly in the problem.)

  8. Answer: yes, viable.
    Checking both inequalities: 90 + 50 = 140, which is ≤ 150, so the total-use inequality is satisfied. Also, 90 ≥ 70, so the indoor-minimum inequality is satisfied. Both constraints hold, so this combination is viable.

  9. Answer: not viable; fails the indoor-minimum constraint.
    Checking both inequalities: 60 + 80 = 140, which is ≤ 150, so the total-use inequality is satisfied. However, 60 is not ≥ 70, so the indoor-minimum inequality fails. Even though the household's total use is under the 150-gallon limit, using only 60 gallons indoors does not meet the basic health requirement, so this combination is not viable.

Differentiation

Support

Provide the equation V = 480 − 6d already written for problem 1, and walk through solving it for d = 30 as a class example before releasing problems 2 through 6. For problems 7 through 9, provide a printed checklist with the two inequalities already written so students only need to substitute and check.

Extension

Ask students to find the decline rate, in million gallons per day, that would let the reservoir last 30 additional days after day 20 before reaching 300 million gallons (answer: 2 million gallons per day), and to explain what a decline rate of exactly 0 million gallons per day would mean for the reservoir's volume equation.

ELL supports

Provide a labeled number line showing "day 0" through "day 40," with the reservoir's critical minimum marked, so students can visualize the timeline before solving the equations algebraically. For problems 7 through 9, provide a simple two-column checklist labeled "Total use ≤ 150?" and "Indoor use ≥ 70?" for students to mark yes or no for each household.