{
  "version": 1,
  "topic": "CE09",
  "title": "Barrels cubic feet and gallons in cement calculations",
  "createdAt": "2026-10-24",
  "case": {
    "title": "One volume on three unit bases",
    "premise": "Petroleum barrel 42 US gallons; US gallon 231 in³; cubic foot 1,728 in³. Density is irrelevant to these conversions.",
    "formula": "US gal = 42 bbl; ft³ = bbl × 42 × 231/1728",
    "variables": "These are geometric volume units, not reservoir-to-stock-tank state conversions.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Volume (bbl) = 10",
        "explanation": "Petroleum barrel 42 US gallons; US gallon 231 in³; cubic foot 1,728 in³. Density is irrelevant to these conversions."
      },
      {
        "label": "Select the relation",
        "calculation": "US gal = 42 bbl; ft³ = bbl × 42 × 231/1728",
        "explanation": "These are geometric volume units, not reservoir-to-stock-tank state conversions."
      },
      {
        "label": "Substitute the values",
        "calculation": "10 × 42 = 420 US gal; 420 × 231/1728 = 56.1458333 ft³",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Volume (US gal) = 420; Volume (ft³) = 56.1458333",
        "explanation": "The three numbers represent the same physical volume in different units."
      }
    ],
    "result": "Volume (US gal) = 420; Volume (ft³) = 56.1458333",
    "interpretation": "The three numbers represent the same physical volume in different units.",
    "control": "Volume (bbl)",
    "columns": [
      "Volume (bbl)",
      "Volume (US gal)",
      "Volume (ft³)"
    ],
    "rows": [
      [
        1,
        42,
        5.614583333333333
      ],
      [
        5,
        210,
        28.072916666666664
      ],
      [
        10,
        420,
        56.14583333333333
      ],
      [
        50,
        2100,
        280.72916666666663
      ],
      [
        100,
        4200,
        561.4583333333333
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Volume (bbl) = 1. Volume (US gal): 42; Volume (ft³): 5.61458333.",
      "Volume (bbl) = 5. Volume (US gal): 210; Volume (ft³): 28.0729167.",
      "Volume (bbl) = 10. Volume (US gal): 420; Volume (ft³): 56.1458333.",
      "Volume (bbl) = 50. Volume (US gal): 2100; Volume (ft³): 280.729167.",
      "Volume (bbl) = 100. Volume (US gal): 4200; Volume (ft³): 561.458333."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Volume (US gal) without rounding intermediate values.",
        "answer": "10 × 42 = 420 US gal; 420 × 231/1728 = 56.1458333 ft³. Volume (US gal) = 420. The three numbers represent the same physical volume in different units."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Volume (bbl) changes from 1 to 100. Other controls remain as stated in the premise. Volume (US gal) changes from 42 to 4200."
      },
      {
        "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. These are geometric volume units, not reservoir-to-stock-tank state conversions. The three numbers represent the same physical volume in different units."
      }
    ]
  },
  "recorded": null
}