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Math problem set Grade 12 Math

Thornfield Wastewater Plant: BOD Removal and Exponential Decay

A grade 12 problem set applying percent removal and exponential decay functions to BOD data from a fictional wastewater treatment plant.

Objectives

  1. Students will calculate percent removal efficiency from influent and effluent concentrations.
  2. Students will construct and use an exponential function to model repeated fixed-percent removal across multiple treatment stages.

Vocabulary

Biochemical oxygen demand (BOD)
Biochemical oxygen demand, or BOD, measures how much oxygen tiny microbes will use up as they eat the waste in water. Dirtier water has a higher BOD.
Influent
Influent is the incoming dirty water that arrives at a treatment plant, ready to be cleaned. Operators test it to see how much work the plant has to do.
Effluent
Effluent is the cleaned water that leaves a treatment plant at the end of the process. It is tested carefully before being released to a river, lake, or the ocean.

Also: removal efficiency, treatment train

Teacher overview

Thornfield Wastewater Treatment Plant (a fictional plant used for this problem set) removes biochemical oxygen demand, or BOD, from sewage before releasing treated water back into a river. Federal secondary treatment standards require a monthly-average effluent BOD of 30 mg/L or less and at least 85 percent removal, and raw domestic sewage typically enters a plant with a BOD of about 100 to 300 mg/L, with medium-strength sewage near 200 mg/L. This problem set starts with Thornfield's monthly removal numbers, then models how BOD drops stage by stage through the plant's treatment train as an exponential function.

A common misconception in the second half of this set is assuming that if each treatment stage removes 40 percent of the BOD, then four stages must remove a full 160 percent, or otherwise "use up" all the BOD in a fixed number of steps. Removal by a constant percentage is multiplicative, not additive: each stage removes 40 percent of whatever BOD is left entering that stage, not 40 percent of the original amount, which is exactly why problems 6 and 7 ask students to check their intuition against the actual exponential function.

Thornfield Wastewater Plant: BOD Removal and Exponential Decay — Grade 12 Math
Name: ____________________________ Date: ______________

Problems

  1. This month, at the fictional Thornfield plant, raw influent BOD tested at 250 mg/L, within the typical range of about 100 to 300 mg/L for raw domestic sewage, and the plant's effluent BOD tested at 22 mg/L. Using the removal efficiency formula, Removal (%) = (influent − effluent) ÷ influent × 100, calculate Thornfield's percent removal efficiency for this month. Round to the nearest tenth of a percent.

  2. Does this month's effluent BOD of 22 mg/L comply with the federal secondary treatment standard of 30 mg/L or less? By how many mg/L does it clear the standard?

  3. Federal secondary treatment standards also require at least 85 percent removal. Compare Thornfield's removal efficiency from problem 1 with this 85 percent minimum: is Thornfield above or below it, and by how many percentage points?

  4. Thornfield's engineers model how BOD drops through the plant's treatment train as a sequence of stages, where each stage removes a fixed 40 percent of the BOD entering it (so 60 percent of each stage's incoming BOD carries forward to the next stage). Write a function B(n) that gives the BOD, in mg/L, remaining after n treatment stages, starting from the raw influent of 250 mg/L.

  5. Use your function from problem 4 to calculate the BOD remaining after 2 stages and after 4 stages. Round each answer to the nearest tenth of a mg/L.

  6. Is 4 stages enough to bring BOD down to the 30 mg/L secondary treatment standard? If not, calculate the BOD remaining after 5 stages, B(5), and state whether 5 stages meets the standard.

  7. Solving 250 × 0.60ⁿ = 30 exactly gives a value of n around 4.15 stages. Explain why Thornfield's engineers must round this number up to a whole number of real treatment stages rather than rounding down or building "4.15 stages," and state the minimum whole number of stages the plant actually needs based on your answer to problem 6.

  8. A second, smaller package plant in Thornfield's district starts with a lower raw influent BOD of 200 mg/L, also within the typical range, and uses the same 40 percent-per-stage removal rate. After 3 stages, is this plant's BOD below the 30 mg/L secondary treatment standard? Show the calculation that supports your answer.

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: 91.2 percent.
    Removal = (250 − 22) ÷ 250 × 100 = 228 ÷ 250 × 100 = 91.2 percent.

  2. Answer: 8 mg/L margin.
    22 mg/L is below the 30 mg/L standard, complying with the requirement. Margin = 30 − 22 = 8 mg/L.

  3. Answer: 6.2 percentage points above.
    91.2 percent (Thornfield's removal) − 85 percent (required minimum) = 6.2 percentage points above the 85 percent minimum.

  4. Answer: B(n) = 250 × 0.60ⁿ.
    Each stage retains 60 percent of the BOD entering it (100 percent minus the 40 percent removed), so after n stages the starting 250 mg/L has been multiplied by 0.60 a total of n times.

  5. Answer: 90.0 mg/L after 2 stages; 32.4 mg/L after 4 stages.
    B(2) = 250 × 0.60² = 250 × 0.36 = 90.0 mg/L. B(4) = 250 × 0.60⁴ = 250 × 0.1296 = 32.4 mg/L.

  6. Answer: No; B(5) = 19.4 mg/L, which does meet the standard.
    B(4) = 32.4 mg/L still exceeds the 30 mg/L standard. B(5) = 250 × 0.60⁵ = 250 × 0.07776 = 19.44 mg/L, which rounds to 19.4 mg/L and is below the 30 mg/L standard.

  7. Answer: 5 stages.
    A real treatment train can only be built with a whole number of physical stages, so a fractional value like 4.15 has no physical meaning. Rounding down to 4 stages leaves the BOD at 32.4 mg/L, still above the 30 mg/L standard and out of compliance, so the plant must round up to 5 stages, the minimum whole number of stages that actually satisfies the requirement.

  8. Answer: No, not yet below the standard.
    B(3) = 200 × 0.60³ = 200 × 0.216 = 43.2 mg/L, which is still above the 30 mg/L secondary treatment standard, so this plant is not yet in compliance after 3 stages and would need at least one more stage.

Differentiation

Support (below grade level)

Provide a partially completed table of B(n) values for n = 0 through 5, with the multiplication for n = 0 and n = 1 already shown, so students can extend the pattern for problems 5 and 6 rather than building the function from scratch. Allow calculators for all exponent calculations.

Extension (above grade level)

Ask students to solve 250 × 0.60ⁿ = 30 exactly for n using logarithms (n = log(30 ÷ 250) ÷ log(0.60)), confirm their answer is close to 4.15, and explain how this exact algebraic method connects to the guess-and-check approach used in problems 5 and 6.

ELL supports

Pre-teach "influent," "effluent," and "treatment train" using a simple flow diagram showing sewage entering on the left, passing through labeled stage boxes, and exiting as treated water on the right, with the BOD number shrinking at each stage. Allow students to show their exponential function work as a labeled table alongside the equation rather than in a full paragraph.