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Worksheet Grade 12 Math

Aspenmill Dam Releases: A Piecewise Rule and Its Domain

A grade 12 worksheet where students evaluate and invert a three-branch piecewise release rule for a dam, state its domain, and convert cfs into acre-feet.

Objectives

  1. Students will evaluate a three-branch piecewise function at given inputs.
  2. Students will solve each branch of a piecewise function for a stated output value.
  3. Students will state a function's domain in terms of a physical limit and convert flow into acre-feet.

Vocabulary

Acre-foot
The amount of water it takes to cover one acre of land, about the size of a football field, one foot deep. Western states use acre-feet to measure large amounts of water.
Outlet works
The pipes, gates, and valves built through a dam that let operators release water downstream on purpose and control the reservoir level.

Also: piecewise function, domain, cubic feet per second

Teacher overview

A dam's operating rule tells the operator how much water must go downstream at any reservoir level, and because the obligations change as the reservoir falls, the rule is commonly written in branches rather than as a single formula. This worksheet uses the fictional Aspenmill Dam on the Drayfen River, whose required release drops in one pattern between the full pool and an intermediate elevation and in a gentler pattern below that. Students evaluate the rule, solve each branch backward for a target release, and state the elevations over which the rule means anything at all. The last three items convert a required release of 190 cfs into acre-feet, which is the unit a reservoir accounting sheet actually uses.

Aspenmill Dam Releases: A Piecewise Rule and Its Domain — Grade 12 Math
Name: ____________________________ Date: ______________

Student Worksheet

The fictional Aspenmill Dam releases water into the Drayfen River according to the rule below. In it, E is the reservoir elevation in feet and Q is the required release in cubic feet per second, written cfs.

Branch 1: when E is 640 feet or higher, Q = 250.

Branch 2: when E is at least 610 feet but below 640 feet, Q = 250 - 5(640 - E).

Branch 3: when E is at least 590 feet but below 610 feet, Q = 100 - 2(610 - E).

The lowest outlet in the dam sits at elevation 590 feet, and water spills over the crest at elevation 660 feet.

  1. Find the required release when the reservoir stands at 652 feet.
  2. Find the required release when the reservoir stands at 628 feet.
  3. Find the required release when the reservoir stands at exactly 610 feet.
  4. Find the required release when the reservoir stands at 598 feet.
  5. Solve for the elevation at which the required release is 175 cfs.
  6. Solve for the elevation at which the required release is 80 cfs.
  7. State the domain of this release rule as an interval of elevations, in feet.
  8. Explain in one or two sentences why the rule assigns no release at all to elevations below 590 feet.
  9. Convert a release of 190 cfs into gallons per minute, using 448.8 gallons per minute per cfs.
  10. Using your answer to question 9, how many gallons does the fictional dam release in one full day of 1,440 minutes?
  11. The reservoir holds steady at 628 feet for 6 days. Using 325,851 gallons per acre-foot, how many acre-feet are released over those 6 days? Round to the nearest acre-foot.
  12. Explain in two or three sentences why an operating rule like this one is written as a piecewise function instead of as a single equation covering every elevation.

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: 250 cfs.
    An elevation of 652 feet is above 640 feet, so Branch 1 applies and the required release is the flat 250 cfs.

  2. Answer: 190 cfs.
    Branch 2 applies at 628 feet: Q = 250 - 5(640 - 628) = 250 - 5(12) = 250 - 60 = 190 cfs.

  3. Answer: 100 cfs.
    At exactly 610 feet, Branch 2 gives Q = 250 - 5(640 - 610) = 250 - 150 = 100 cfs. Branch 3 gives the same value there, so the two pieces meet without a jump.

  4. Answer: 76 cfs.
    Branch 3 applies at 598 feet: Q = 100 - 2(610 - 598) = 100 - 2(12) = 100 - 24 = 76 cfs.

  5. Answer: 625 feet.
    Setting 250 - 5(640 - E) = 175 gives 5(640 - E) = 75, so 640 - E = 15 and E = 625 feet, which falls inside Branch 2's interval.

  6. Answer: 600 feet.
    Setting 100 - 2(610 - E) = 80 gives 2(610 - E) = 20, so 610 - E = 10 and E = 600 feet, which falls inside Branch 3's interval.

  7. Answer: From 590 feet to 660 feet.
    The rule is defined from the lowest outlet at 590 feet up to the spill crest at 660 feet. Accept the interval written with inequality symbols or in interval notation.

  8. Answer: The outlet works cannot draw water below 590 feet.
    The lowest opening in the dam sits at that elevation, so there is no physical way to make a release once the surface drops below it, and the function has nothing to assign.

  9. Answer: 85,272 gpm.
    Multiplying 190 cfs by 448.8 gallons per minute per cfs gives 85,272 gallons per minute.

  10. Answer: 122,791,680 gallons.
    Multiplying 85,272 gallons per minute by 1,440 minutes gives 122,791,680 gallons in one day.

  11. Answer: 2,261 acre-feet.
    Six days at 122,791,680 gallons per day is 736,750,080 gallons. Dividing by 325,851 gallons per acre-foot gives about 2,261.0, which rounds to 2,261 acre-feet.

  12. Answer: Different obligations govern at different reservoir levels.
    Near full pool the fictional dam owes a fixed downstream flow, while at lower levels the release is cut back at two different rates to protect the remaining storage. One straight line cannot produce a flat stretch and two different slopes, so the rule needs a separate expression on each interval.

Differentiation

Support

Give these students a graph of the fictional Aspenmill rule already drawn on labeled axes, with the two breakpoints marked, so items 1 through 6 become reading the graph and confirming with arithmetic. Provide a worked example of one branch solved backward on a neutral function, and supply the conversion chain for items 9 through 11 as a labeled sequence of three boxes.

Extension

Ask students to add a fourth branch to the fictional rule for flood operations, in which the release rises by 40 cfs for every foot the reservoir stands above 655 feet, and to write the whole four-branch function. Then have them check whether the new branch meets Branch 1 without a jump at 655 feet, and describe what a jump would mean for the river below the dam.

ELL supports

Pre-teach "piecewise," "domain," "outlet works," and "acre-foot" with a labeled cross-section of a dam showing the spill crest, the lowest outlet, and the two breakpoint elevations. Provide sentence frames for items 8 and 12, such as "The rule stops at 590 feet because ___," and let students answer item 12 with a labeled sketch of the three branches before writing.

Extensions

Students can graph the fictional Aspenmill rule on labeled axes and mark the two breakpoints, then describe in writing whether the function is continuous and where its slope changes. A second follow-up asks students to look up how a real reservoir near them publishes its operating curve and to compare the number of branches in that rule with the three used here.