{
  "version": 1,
  "topic": "GM08",
  "title": "Explore the square root of time in Carter leakoff",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Fixed-area Carter loss with spurt separated",
    "premise": "Effective area 20,000 ft², CL = 0.003 ft/√min, Sp = 0.02 ft. All area is exposed at once.",
    "formula": "VCarter = 2ALCL√t; Vtotal = VCarter + ALSp",
    "variables": "Exposure in minutes; no extra face factor is added to the declared effective area.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Exposure time (min) = 60",
        "explanation": "Effective area 20,000 ft², CL = 0.003 ft/√min, Sp = 0.02 ft. All area is exposed at once."
      },
      {
        "label": "Select the relation",
        "calculation": "VCarter = 2ALCL√t; Vtotal = VCarter + ALSp",
        "explanation": "Exposure in minutes; no extra face factor is added to the declared effective area."
      },
      {
        "label": "Substitute the values",
        "calculation": "At 60 min: 2 × 20,000 × 0.003 × √60 = 929.516003 ft³; spurt = 20,000 × 0.02 = 400 ft³; total = 1,329.516003 ft³",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Carter component (ft³) = 929.516003; Total with spurt (ft³) = 1329.516; Carter ratio to 60 min = 1",
        "explanation": "Only the Carter component follows the pure square-root ratio when spurt is nonzero."
      }
    ],
    "result": "Carter component (ft³) = 929.516003; Total with spurt (ft³) = 1329.516; Carter ratio to 60 min = 1",
    "interpretation": "Only the Carter component follows the pure square-root ratio when spurt is nonzero.",
    "control": "Exposure time (min)",
    "columns": [
      "Exposure time (min)",
      "Carter component (ft³)",
      "Total with spurt (ft³)",
      "Carter ratio to 60 min"
    ],
    "rows": [
      [
        15,
        464.75800154489,
        864.75800154489,
        0.5
      ],
      [
        30,
        657.2670690061993,
        1057.2670690061993,
        0.7071067811865476
      ],
      [
        60,
        929.51600308978,
        1329.51600308978,
        1
      ],
      [
        120,
        1314.5341380123987,
        1714.534138012399,
        1.4142135623730951
      ],
      [
        240,
        1859.03200617956,
        2259.03200617956,
        2
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Exposure time (min) = 15. Carter component (ft³): 464.758002; Total with spurt (ft³): 864.758002; Carter ratio to 60 min: 0.5.",
      "Exposure time (min) = 30. Carter component (ft³): 657.267069; Total with spurt (ft³): 1057.26707; Carter ratio to 60 min: 0.707106781.",
      "Exposure time (min) = 60. Carter component (ft³): 929.516003; Total with spurt (ft³): 1329.516; Carter ratio to 60 min: 1.",
      "Exposure time (min) = 120. Carter component (ft³): 1314.53414; Total with spurt (ft³): 1714.53414; Carter ratio to 60 min: 1.41421356.",
      "Exposure time (min) = 240. Carter component (ft³): 1859.03201; Total with spurt (ft³): 2259.03201; Carter ratio to 60 min: 2."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Carter component (ft³) without rounding intermediate values.",
        "answer": "At 60 min: 2 × 20,000 × 0.003 × √60 = 929.516003 ft³; spurt = 20,000 × 0.02 = 400 ft³; total = 1,329.516003 ft³. Carter component (ft³) = 929.516003. Only the Carter component follows the pure square-root ratio when spurt is nonzero."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Exposure time (min) changes from 15 to 240. Other controls remain as stated in the premise. Carter component (ft³) changes from 464.758002 to 1859.03201."
      },
      {
        "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. Exposure in minutes; no extra face factor is added to the declared effective area. Only the Carter component follows the pure square-root ratio when spurt is nonzero."
      }
    ]
  },
  "recorded": null
}