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Worksheet Grade 12 Math

Briarcliff Water Plant: Chained Unit Conversions and Chlorine Feed

A grade 12 worksheet where students chain MGD, gpm, and cfs conversions and apply the operator pounds formula to set a chlorine feed rate at a fictional treatment plant.

Objectives

  1. Students will convert a treatment plant's flow rate through a chain of standard water-industry units, including MGD, gpm, and cfs.
  2. Students will apply the pounds formula to calculate and constrain a chemical feed rate using unit analysis.

Vocabulary

Key words: million gallons per day (MGD), cubic feet per second (cfs), pounds formula, chemical feed rate

Teacher overview

Briarcliff (a fictional town used for this worksheet) operates a water treatment plant that produces a steady average flow of 2.4 million gallons per day (MGD). Operator-certification exams routinely ask candidates to move that single flow figure through several different units without losing track of what each number represents, because a plant's flow meters, pump curves, and chemical-feed calculations are rarely reported in the same unit twice. This worksheet chains together the plant's flow rate in MGD, gallons per minute (gpm), and cubic feet per second (cfs), then uses the same flow figure inside the pounds formula to size a chlorine feed rate.

A common misconception in this kind of problem is treating each conversion as an isolated fact to memorize rather than a rate that can be chained: students sometimes convert MGD to cfs correctly, then multiply by the wrong factor when converting that cfs figure back to gpm, because they forgot which conversion factor belongs to which unit pair. Watch for that error in items 2 and 3, where the chained answer should land close to, but not always identical to, the direct conversion in item 1, and the reason for the small gap is exactly what item 10 asks students to explain.

Briarcliff Water Plant: Chained Unit Conversions and Chlorine Feed — Grade 12 Math
Name: ____________________________ Date: ______________

Student Worksheet

All problems in this fictional scenario refer to the Briarcliff plant's average daily flow of 2.4 MGD unless a second facility is named. Use these standard conversions as needed: 1 MGD = 694 gpm; 1 MGD = 1.547 cfs; 1 cfs = 448.8 gpm; 1 cubic foot = 7.48 gallons; 1 gallon of water weighs 8.34 pounds; 1 ton = 2,000 pounds. The pounds formula states: loading (lbs/day) = concentration (mg/L) × flow (MGD) × 8.34.

  1. Convert the plant's flow of 2.4 MGD directly to gallons per minute (gpm). Round to the nearest whole gpm.

  2. Convert the plant's flow of 2.4 MGD directly to cubic feet per second (cfs). Round to the nearest hundredth of a cfs.

  3. Convert your rounded answer from problem 2 back to gpm using 1 cfs = 448.8 gpm. Round to the nearest whole gpm, and note whether it matches your answer to problem 1 exactly.

  4. Convert 2.4 MGD to gallons per day, then use that figure to calculate the total weight of water, in pounds, the plant produces in one day.

  5. Convert your answer from problem 4 into tons of water per day.

  6. Convert the plant's daily flow of 2.4 MGD into cubic feet per day, using 1 cubic foot = 7.48 gallons. Round to the nearest whole cubic foot.

  7. Briarcliff's operators dose chlorine at a target concentration of 2.5 mg/L. Using the pounds formula and the plant's 2.4 MGD flow, calculate the required chlorine feed rate in pounds per day. Round to the nearest tenth of a pound.

  8. Briarcliff's state permit caps chlorine feed at this plant at 60 pounds per day. At the current 2.4 MGD flow, what is the maximum chlorine concentration, in mg/L, the plant may dose without exceeding that cap? Round down to the nearest hundredth of a mg/L so the feed stays within the cap.

  9. Eastview Pump Station, a second fictional facility in the same district, runs at 1.6 MGD and doses chlorine at 3.0 mg/L. Calculate Eastview's chlorine feed rate in pounds per day, then find the combined chlorine feed rate, in pounds per day, for Briarcliff and Eastview together.

  10. Explain why the chained conversion in problem 3 (MGD to cfs to gpm) can land a few gpm away from the direct conversion in problem 1 (MGD straight to gpm), even though both start from the same 2.4 MGD flow. Which method would you recommend an operator use when precision matters, and why?

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: 1,666 gpm.
    2.4 MGD × 694 gpm/MGD = 1,665.6 gpm, which rounds to 1,666 gpm.

  2. Answer: 3.71 cfs.
    2.4 MGD × 1.547 cfs/MGD = 3.7128 cfs, which rounds to 3.71 cfs.

  3. Answer: 1,665 gpm.
    3.71 cfs × 448.8 gpm/cfs = 1,665.05 gpm, which rounds to 1,665 gpm. This is 1 gpm less than problem 1's answer of 1,666 gpm, because problem 3 chains through the rounded 3.71 cfs value instead of using the full unrounded flow.

  4. Answer: 20,016,000 lb/day.
    2.4 MGD = 2,400,000 gallons/day. 2,400,000 gal/day × 8.34 lb/gal = 20,016,000 lb/day.

  5. Answer: 10,008 tons/day.
    20,016,000 lb/day ÷ 2,000 lb/ton = 10,008 tons/day.

  6. Answer: 320,856 cu ft/day.
    2,400,000 gal/day ÷ 7.48 gal/cu ft = 320,855.61 cu ft/day, which rounds to 320,856 cu ft/day.

  7. Answer: 50.0 lb/day.
    Loading = 2.5 mg/L × 2.4 MGD × 8.34 = 50.04 lb/day, which rounds to 50.0 lb/day.

  8. Answer: 2.99 mg/L.
    Maximum concentration = 60 lb/day ÷ (2.4 MGD × 8.34) = 60 ÷ 20.016 = 2.9976 mg/L, which rounds down to 2.99 mg/L. Rounding up to 3.00 mg/L would give 3.00 × 2.4 × 8.34 = 60.05 lb/day, slightly over the cap, while 2.99 mg/L gives 59.85 lb/day.

  9. Answer: 90.1 lb/day combined.
    Eastview: 3.0 mg/L × 1.6 MGD × 8.34 = 40.032 lb/day, which rounds to 40.0 lb/day. Combined with Briarcliff's 50.04 lb/day from problem 7: 50.04 + 40.032 = 90.072 lb/day, which rounds to 90.1 lb/day.

  10. Answer: The gap comes from rounding the intermediate cfs value before converting it back to gpm.
    Problem 1 multiplies the full 2.4 MGD flow by 694 in one step, so no rounding happens until the final answer. Problem 3 first rounds the flow to 3.71 cfs and then multiplies that already-rounded number by 448.8, so the rounding error from step one carries into step two. An operator who needs the most precise possible answer should convert directly between the two units whenever a direct conversion factor is available, saving chained conversions for situations where no direct factor exists.

Differentiation

Support (below grade level)

Provide a printed conversion reference card listing all six conversion facts from the worksheet header, and work through problem 1 as a class before students attempt problems 2 and 3 independently. For the pounds-formula problems (7 through 9), give students the formula already set up with blanks for the three known values, so they only need to multiply.

Extension (above grade level)

Ask students to derive the exact relationship between the 694 gpm/MGD and 1.547 cfs/MGD conversion factors by showing that 694 gpm ÷ 448.8 gpm/cfs should equal approximately 1.547 cfs, and to explain in writing why very small differences appear when working from rounded industry-standard constants instead of exact fractions.

ELL supports

Pre-teach the abbreviations MGD, gpm, and cfs as vocabulary cards paired with a simple diagram of a flow meter, a spinning pump wheel, and a stopwatch. Allow students to keep the conversion facts visible at all times and to show their unit labels at every step of their work, since tracking units in writing supports both the math and the English vocabulary at once.