{
  "version": 1,
  "topic": "CE03",
  "title": "Cement slurry yield versus slurry density",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Sacks from a fixed slurry volume",
    "premise": "Planned slurry demand 309.25125 ft³. Yield changes; volume and recipe interpretation are otherwise fixed.",
    "formula": "theoretical sacks = V/Y; whole sacks = ceil(V/Y)",
    "variables": "V in ft³ and Y in ft³/sk. Different yields represent hypothetical documented recipe choices, not a recipe design.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Yield (ft³/sk) = 1.32",
        "explanation": "Planned slurry demand 309.25125 ft³. Yield changes; volume and recipe interpretation are otherwise fixed."
      },
      {
        "label": "Select the relation",
        "calculation": "theoretical sacks = V/Y; whole sacks = ceil(V/Y)",
        "explanation": "V in ft³ and Y in ft³/sk. Different yields represent hypothetical documented recipe choices, not a recipe design."
      },
      {
        "label": "Substitute the values",
        "calculation": "309.25125/1.32 = 234.28125 sk; ceil = 235 sk",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Theoretical sacks (sk) = 234.28125; Whole sacks (sk) = 235",
        "explanation": "A higher yield reduces sacks for the same slurry volume, but must belong to the actual design."
      }
    ],
    "result": "Theoretical sacks (sk) = 234.28125; Whole sacks (sk) = 235",
    "interpretation": "A higher yield reduces sacks for the same slurry volume, but must belong to the actual design.",
    "control": "Yield (ft³/sk)",
    "columns": [
      "Yield (ft³/sk)",
      "Theoretical sacks (sk)",
      "Whole sacks (sk)"
    ],
    "rows": [
      [
        1,
        309.25125,
        310
      ],
      [
        1.2,
        257.709375,
        258
      ],
      [
        1.32,
        234.28125,
        235
      ],
      [
        1.5,
        206.16750000000002,
        207
      ],
      [
        1.8,
        171.80625,
        172
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Yield (ft³/sk) = 1. Theoretical sacks (sk): 309.25125; Whole sacks (sk): 310.",
      "Yield (ft³/sk) = 1.2. Theoretical sacks (sk): 257.709375; Whole sacks (sk): 258.",
      "Yield (ft³/sk) = 1.32. Theoretical sacks (sk): 234.28125; Whole sacks (sk): 235.",
      "Yield (ft³/sk) = 1.5. Theoretical sacks (sk): 206.1675; Whole sacks (sk): 207.",
      "Yield (ft³/sk) = 1.8. Theoretical sacks (sk): 171.80625; Whole sacks (sk): 172."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Theoretical sacks (sk) without rounding intermediate values.",
        "answer": "309.25125/1.32 = 234.28125 sk; ceil = 235 sk. Theoretical sacks (sk) = 234.28125. A higher yield reduces sacks for the same slurry volume, but must belong to the actual design."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Yield (ft³/sk) changes from 1 to 1.8. Other controls remain as stated in the premise. Theoretical sacks (sk) changes from 309.25125 to 171.80625."
      },
      {
        "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. V in ft³ and Y in ft³/sk. Different yields represent hypothetical documented recipe choices, not a recipe design. A higher yield reduces sacks for the same slurry volume, but must belong to the actual design."
      }
    ]
  },
  "recorded": null
}