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Learn through experiments

Less pollution added does not mean none remains

How do accumulation and removal affect intervention outcomes?

Entering midway? Revisit A full reservoir can hide a shortage for the preceding worked example, or start here and use the explanation below.

1 · Try

Start with a concrete question

Original synthetic scenario. A fictional pollutant starts at 20 units. Each period removes a fraction of the starting stock, then adds new emissions. A policy reduces emissions from a chosen period.

Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.

Go directly to the experiment · Review basic mathematics

2 · Predict

What do you expect?

If emissions fall by half, will the accumulated stock immediately fall by half?

3 · Experiment

Change one thing. Inspect what happens.

  1. Observe accumulation with no intervention and explain why removal grows with stock.
  2. Reduce emissions by half from period seven; inspect the first affected update.
  3. Compare early and late intervention with the same reduction.
  4. Set removal to zero and explain why ending emissions stops growth but does not clear the accumulated stock.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Final pollutant stock29.492 units
Total added90 units
Total removed80.508 units

Compare the current results

Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.
042.1357
  • Period 1 pollutant stock (units)26
    Period 1 pollutant stock (units): 26Horizontal bar from zero to 26, on the shared zero to 42.1357 numeric axis.
  • Period 2 pollutant stock (units)30.8
    Period 2 pollutant stock (units): 30.8Horizontal bar from zero to 30.8, on the shared zero to 42.1357 numeric axis.
  • Period 3 pollutant stock (units)34.64
    Period 3 pollutant stock (units): 34.64Horizontal bar from zero to 34.64, on the shared zero to 42.1357 numeric axis.
  • Period 4 pollutant stock (units)37.712
    Period 4 pollutant stock (units): 37.712Horizontal bar from zero to 37.712, on the shared zero to 42.1357 numeric axis.
  • Period 5 pollutant stock (units)40.1696
    Period 5 pollutant stock (units): 40.1696Horizontal bar from zero to 40.1696, on the shared zero to 42.1357 numeric axis.
  • Period 6 pollutant stock (units)42.1357
    Period 6 pollutant stock (units): 42.1357Horizontal bar from zero to 42.1357, on the shared zero to 42.1357 numeric axis.
  • Period 7 pollutant stock (units)38.7085
    Period 7 pollutant stock (units): 38.7085Horizontal bar from zero to 38.7085, on the shared zero to 42.1357 numeric axis.
  • Period 8 pollutant stock (units)35.9668
    Period 8 pollutant stock (units): 35.9668Horizontal bar from zero to 35.9668, on the shared zero to 42.1357 numeric axis.
  • Period 9 pollutant stock (units)33.7735
    Period 9 pollutant stock (units): 33.7735Horizontal bar from zero to 33.7735, on the shared zero to 42.1357 numeric axis.
  • Period 10 pollutant stock (units)32.0188
    Period 10 pollutant stock (units): 32.0188Horizontal bar from zero to 32.0188, on the shared zero to 42.1357 numeric axis.
  • Period 11 pollutant stock (units)30.615
    Period 11 pollutant stock (units): 30.615Horizontal bar from zero to 30.615, on the shared zero to 42.1357 numeric axis.
  • Period 12 pollutant stock (units)29.492
    Period 12 pollutant stock (units): 29.492Horizontal bar from zero to 29.492, on the shared zero to 42.1357 numeric axis.

Follow the calculation

  1. Period 1: start 20.000, remove 4.000 (20% of starting stock), add 10.000, next 26.000.
  2. Period 2: start 26.000, remove 5.200 (20% of starting stock), add 10.000, next 30.800.
  3. Period 3: start 30.800, remove 6.160 (20% of starting stock), add 10.000, next 34.640.
  4. Period 4: start 34.640, remove 6.928 (20% of starting stock), add 10.000, next 37.712.
  5. Period 5: start 37.712, remove 7.542 (20% of starting stock), add 10.000, next 40.170.
  6. Period 6: start 40.170, remove 8.034 (20% of starting stock), add 10.000, next 42.136.
  7. Period 7: start 42.136, remove 8.427 (20% of starting stock), add 5.000, next 38.709.
  8. Period 8: start 38.709, remove 7.742 (20% of starting stock), add 5.000, next 35.967.
  9. Period 9: start 35.967, remove 7.193 (20% of starting stock), add 5.000, next 33.773.
  10. Period 10: start 33.773, remove 6.755 (20% of starting stock), add 5.000, next 32.019.
  11. Period 11: start 32.019, remove 6.404 (20% of starting stock), add 5.000, next 30.615.
  12. Period 12: start 30.615, remove 6.123 (20% of starting stock), add 5.000, next 29.492.

Reducing new emissions does not instantly remove accumulated pollution. Natural removal, starting stock and timing determine the remaining burden.

Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.

4 · Understand

Explain the mechanism

What the experiment explains

Original synthetic scenario. A fictional pollutant starts at 20 units. Each period removes a fraction of the starting stock, then adds new emissions. A policy reduces emissions from a chosen period.

A flow changes a stock. Cutting the incoming flow is different from removing what has already accumulated.

At defaults before intervention, the first update is 20−0.2×20+10=26. The next is 26−5.2+10=30.8. Removal is a fraction of current stock, not a constant amount.

For constant emissions and positive removal fraction, equilibrium is emission/removal. Changing emissions changes the eventual level, but the existing stock adjusts over time.

Read the relationship

Next stock = current stock × (1−removal fraction) + new emissions. Removed amount uses starting stock; all amounts are pollutant units.

Inspect the evidence

The numerical trace is recalculated from your current settings. Compare a controlled pair of trials before changing several inputs together. Inspect intermediate quantities as well as final metrics; state which assumptions make the comparison meaningful.

Boundaries of the model

  • One well-mixed stock, constant fractional removal and no chemistry or transport.
  • Invented units and parameters; not a forecast for a real pollutant or ecosystem.

Return to the environment and sustainability course and project to connect this mechanism to the broader investigation.

5 · Apply

Try a fresh case

A new stock is 50 units, 10% is removed, then 3 units are added. What is the next stock?

This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.

Take the investigation further

Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.

School investigation

Use counters to distinguish emission flow from accumulated stock.

College investigation

Derive the constant-emission equilibrium and compare analytic and iterative trajectories.

Workplace investigation

Define an intervention evaluation that tracks both incoming emissions and remaining burden over time.

6 · Continue

Keep the evidence, choose your next step

Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.