{
  "version": 1,
  "topic": "DC08",
  "title": "Explore the same decline curve on monthly and annual time bases",
  "createdAt": "2026-10-24",
  "case": {
    "title": "One exponential curve on equivalent time bases",
    "premise": "Physical elapsed time 365 days, qi = 1,000 STB/day and D = 0.001/day. A model year is 365 days and a model month is one twelfth of it.",
    "formula": "Dnew = Dday × days per unit; tnew = 365/(days per unit)",
    "variables": "The time basis changes, while Dt and the physical rate remain invariant. The first column is days per display unit.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Days per display unit = 30.4166667",
        "explanation": "Physical elapsed time 365 days, qi = 1,000 STB/day and D = 0.001/day. A model year is 365 days and a model month is one twelfth of it."
      },
      {
        "label": "Select the relation",
        "calculation": "Dnew = Dday × days per unit; tnew = 365/(days per unit)",
        "explanation": "The time basis changes, while Dt and the physical rate remain invariant. The first column is days per display unit."
      },
      {
        "label": "Substitute the values",
        "calculation": "Dmonth = 0.001 × 365/12; t = 12 months; Dmonth × 12 = 0.365; q = 1,000 exp(−0.365)",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Nominal D (1/display unit) = 0.0304166667; Elapsed display units = 12; Endpoint rate (STB/day) = 694.196651",
        "explanation": "Changing axis units together with the coefficient must not change the physical endpoint rate."
      }
    ],
    "result": "Nominal D (1/display unit) = 0.0304166667; Elapsed display units = 12; Endpoint rate (STB/day) = 694.196651",
    "interpretation": "Changing axis units together with the coefficient must not change the physical endpoint rate.",
    "control": "Days per display unit",
    "columns": [
      "Days per display unit",
      "Nominal D (1/display unit)",
      "Elapsed display units",
      "Endpoint rate (STB/day)"
    ],
    "rows": [
      [
        1,
        0.001,
        365,
        694.1966508779789
      ],
      [
        7,
        0.007,
        52.142857142857146,
        694.1966508779789
      ],
      [
        30.416666666666668,
        0.030416666666666668,
        12,
        694.1966508779789
      ],
      [
        365,
        0.365,
        1,
        694.1966508779789
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Days per display unit = 1. Nominal D (1/display unit): 0.001; Elapsed display units: 365; Endpoint rate (STB/day): 694.196651.",
      "Days per display unit = 7. Nominal D (1/display unit): 0.007; Elapsed display units: 52.1428571; Endpoint rate (STB/day): 694.196651.",
      "Days per display unit = 30.4166667. Nominal D (1/display unit): 0.0304166667; Elapsed display units: 12; Endpoint rate (STB/day): 694.196651.",
      "Days per display unit = 365. Nominal D (1/display unit): 0.365; Elapsed display units: 1; Endpoint rate (STB/day): 694.196651."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Nominal D (1/display unit) without rounding intermediate values.",
        "answer": "Dmonth = 0.001 × 365/12; t = 12 months; Dmonth × 12 = 0.365; q = 1,000 exp(−0.365). Nominal D (1/display unit) = 0.0304166667. Changing axis units together with the coefficient must not change the physical endpoint rate."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Days per display unit changes from 1 to 365. Other controls remain as stated in the premise. Nominal D (1/display unit) changes from 0.001 to 0.365."
      },
      {
        "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. The time basis changes, while Dt and the physical rate remain invariant. The first column is days per display unit. Changing axis units together with the coefficient must not change the physical endpoint rate."
      }
    ]
  },
  "recorded": null
}