{
  "version": 1,
  "topic": "DC05",
  "title": "Why a decline forecast needs an end date and economic assumptions",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Rate, integral and endpoint rectangle",
    "premise": "Exponential qi = 1,000 STB/day, nominal D = 0.001/day. The finite elapsed duration varies.",
    "formula": "q = 1,000 exp(−0.001t); Np = 1,000[1 − exp(−0.001t)]/0.001",
    "variables": "t in days gives rate in STB/day and cumulative in STB. Endpoint rectangle q(t)t is shown only as a deliberately approximate comparison.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Elapsed time (day) = 365",
        "explanation": "Exponential qi = 1,000 STB/day, nominal D = 0.001/day. The finite elapsed duration varies."
      },
      {
        "label": "Select the relation",
        "calculation": "q = 1,000 exp(−0.001t); Np = 1,000[1 − exp(−0.001t)]/0.001",
        "explanation": "t in days gives rate in STB/day and cumulative in STB. Endpoint rectangle q(t)t is shown only as a deliberately approximate comparison."
      },
      {
        "label": "Substitute the values",
        "calculation": "Np(365) = 1,000[1 − exp(−0.365)]/0.001; endpoint rectangle = [1,000 exp(−0.365)] × 365",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Analytical cumulative (STB) = 305803.349; Endpoint rectangle (STB) = 253381.778; Endpoint rate (STB/day) = 694.196651",
        "explanation": "The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon."
      }
    ],
    "result": "Analytical cumulative (STB) = 305803.349; Endpoint rectangle (STB) = 253381.778; Endpoint rate (STB/day) = 694.196651",
    "interpretation": "The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon.",
    "control": "Elapsed time (day)",
    "columns": [
      "Elapsed time (day)",
      "Analytical cumulative (STB)",
      "Endpoint rectangle (STB)",
      "Endpoint rate (STB/day)"
    ],
    "rows": [
      [
        30,
        29554.46645149182,
        29113.366006455246,
        970.4455335485081
      ],
      [
        90,
        86068.81472877182,
        82253.80667441053,
        913.9311852712282
      ],
      [
        365,
        305803.3491220212,
        253381.7775704623,
        694.1966508779789
      ],
      [
        730,
        518091.00990979763,
        351793.5627658478,
        481.90899009020245
      ],
      [
        1825,
        838782.3558702233,
        294222.20053684263,
        161.21764412977677
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Elapsed time (day) = 30. Analytical cumulative (STB): 29554.4665; Endpoint rectangle (STB): 29113.366; Endpoint rate (STB/day): 970.445534.",
      "Elapsed time (day) = 90. Analytical cumulative (STB): 86068.8147; Endpoint rectangle (STB): 82253.8067; Endpoint rate (STB/day): 913.931185.",
      "Elapsed time (day) = 365. Analytical cumulative (STB): 305803.349; Endpoint rectangle (STB): 253381.778; Endpoint rate (STB/day): 694.196651.",
      "Elapsed time (day) = 730. Analytical cumulative (STB): 518091.01; Endpoint rectangle (STB): 351793.563; Endpoint rate (STB/day): 481.90899.",
      "Elapsed time (day) = 1825. Analytical cumulative (STB): 838782.356; Endpoint rectangle (STB): 294222.201; Endpoint rate (STB/day): 161.217644."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Analytical cumulative (STB) without rounding intermediate values.",
        "answer": "Np(365) = 1,000[1 − exp(−0.365)]/0.001; endpoint rectangle = [1,000 exp(−0.365)] × 365. Analytical cumulative (STB) = 305803.349. The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Elapsed time (day) changes from 30 to 1825. Other controls remain as stated in the premise. Analytical cumulative (STB) changes from 29554.4665 to 838782.356."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. t in days gives rate in STB/day and cumulative in STB. Endpoint rectangle q(t)t is shown only as a deliberately approximate comparison. The endpoint rectangle misses earlier higher rates. The analytical cumulative remains conditional on the finite horizon."
      }
    ]
  },
  "recorded": null
}