{
  "index": "2006-A-1",
  "type": "GEO",
  "tag": [
    "GEO",
    "ANA"
  ],
  "difficulty": "",
  "question": "Find the volume of the region of points $(x,y,z)$ such that\n\\[\n(x^2 + y^2 + z^2 + 8)^2 \\leq 36(x^2 + y^2).\n\\]",
  "solution": "We change to cylindrical coordinates, i.e., we put $r = \\sqrt{x^2 + y^2}$.\nThen the given inequality is equivalent to\n\\[\nr^2 + z^2 + 8 \\leq 6r,\n\\]\nor\n\\[\n(r-3)^2 + z^2 \\leq 1.\n\\]\nThis defines a solid of revolution (a solid torus); the area being rotated\nis the disc $(x-3)^2 + z^2 \\leq 1$ in the $xz$-plane. By Pappus's theorem,\nthe volume of this equals the area of this disc, which is $\\pi$, times the\ndistance through which the center of mass is being rotated, which is $(2\\pi)3$.\nThat is, the total volume is $6 \\pi^2$.",
  "vars": [
    "x",
    "y",
    "z",
    "r"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "abscissa",
        "y": "ordinate",
        "z": "applicate",
        "r": "radialst"
      },
      "question": "Find the volume of the region of points $(abscissa,ordinate,applicate)$ such that\n\\[\n(abscissa^2 + ordinate^2 + applicate^2 + 8)^2 \\leq 36(abscissa^2 + ordinate^2).\n\\]",
      "solution": "We change to cylindrical coordinates, i.e., we put $radialst = \\sqrt{abscissa^2 + ordinate^2}$. Then the given inequality is equivalent to\n\\[\nradialst^2 + applicate^2 + 8 \\leq 6\\, radialst,\n\\]\nor\n\\[\n(radialst-3)^2 + applicate^2 \\leq 1.\n\\]\nThis defines a solid of revolution (a solid torus); the area being rotated\nis the disc $(abscissa-3)^2 + applicate^2 \\leq 1$ in the $abscissa\\,applicate$-plane. By Pappus's theorem,\nthe volume of this equals the area of this disc, which is $\\pi$, times the\ndistance through which the center of mass is being rotated, which is $(2\\pi)3$.\nThat is, the total volume is $6 \\pi^2$.}"
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "watermelon",
        "y": "toucanbill",
        "z": "raincloud",
        "r": "mousetrap"
      },
      "question": "Find the volume of the region of points $(watermelon,toucanbill,raincloud)$ such that\n\\[\n(watermelon^2 + toucanbill^2 + raincloud^2 + 8)^2 \\leq 36(watermelon^2 + toucanbill^2).\n\\]",
      "solution": "We change to cylindrical coordinates, i.e., we put $mousetrap = \\sqrt{watermelon^2 + toucanbill^2}$. Then the given inequality is equivalent to\n\\[\nmousetrap^2 + raincloud^2 + 8 \\leq 6mousetrap,\n\\]\nor\n\\[\n(mousetrap-3)^2 + raincloud^2 \\leq 1.\n\\]\nThis defines a solid of revolution (a solid torus); the area being rotated is the disc $(watermelon-3)^2 + raincloud^2 \\leq 1$ in the $watermelon raincloud$-plane. By Pappus's theorem, the volume of this equals the area of this disc, which is $\\pi$, times the distance through which the center of mass is being rotated, which is $(2\\pi)3$. That is, the total volume is $6 \\pi^2$.} يونيو ഒ 영역 (The last part "
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "verticalaxis",
        "y": "fixedvalue",
        "z": "horizontalplane",
        "r": "nonradius"
      },
      "question": "Find the volume of the region of points $(verticalaxis,fixedvalue,horizontalplane)$ such that\n\\[\n(verticalaxis^2 + fixedvalue^2 + horizontalplane^2 + 8)^2 \\leq 36(verticalaxis^2 + fixedvalue^2).\n\\]",
      "solution": "We change to cylindrical coordinates, i.e., we put $nonradius = \\sqrt{verticalaxis^2 + fixedvalue^2}$. Then the given inequality is equivalent to\n\\[\nnonradius^2 + horizontalplane^2 + 8 \\leq 6nonradius,\n\\]\nor\n\\[\n(nonradius-3)^2 + horizontalplane^2 \\leq 1.\n\\]\nThis defines a solid of revolution (a solid torus); the area being rotated is the disc $(verticalaxis-3)^2 + horizontalplane^2 \\leq 1$ in the $verticalaxis horizontalplane$-plane. By Pappus's theorem, the volume of this equals the area of this disc, which is $\\pi$, times the distance through which the center of mass is being rotated, which is $(2\\pi)3$. That is, the total volume is $6 \\pi^2$. "
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "z": "ptbylqmn",
        "r": "vsnckaeu"
      },
      "question": "Find the volume of the region of points $(qzxwvtnp,hjgrksla,ptbylqmn)$ such that\n\\[\n(qzxwvtnp^2 + hjgrksla^2 + ptbylqmn^2 + 8)^2 \\leq 36(qzxwvtnp^2 + hjgrksla^2).\n\\]",
      "solution": "We change to cylindrical coordinates, i.e., we put $vsnckaeu = \\sqrt{qzxwvtnp^2 + hjgrksla^2}$. Then the given inequality is equivalent to\n\\[\nvsnckaeu^2 + ptbylqmn^2 + 8 \\leq 6vsnckaeu,\n\\]\nor\n\\[\n(vsnckaeu-3)^2 + ptbylqmn^2 \\leq 1.\n\\]\nThis defines a solid of revolution (a solid torus); the area being rotated is the disc $(qzxwvtnp-3)^2 + ptbylqmn^2 \\leq 1$ in the $qzxwvtnpptbylqmn$-plane. By Pappus's theorem, the volume of this equals the area of this disc, which is $\\pi$, times the distance through which the center of mass is being rotated, which is $(2\\pi)3$. That is, the total volume is $6 \\pi^2$. "
    },
    "kernel_variant": {
      "question": "Determine the volume of the set of all points $(x,y,z)$ in $\\mathbb R^3$ satisfying\n\\[\n\\bigl(x^{2}+y^{2}+z^{2}+5\\bigr)^{2}\\;\\le\\;25\\,(x^{2}+y^{2}).\n\\]",
      "solution": "Introduce cylindrical coordinates r=√(x²+y²) and keep z unchanged.  The inequality becomes\n\n  (r²+z²+5)² ≤25r²,  r≥0.\n\nSince both sides are non–negative we may take square roots:\n\n  r²+z²+5 ≤5r.\n\nRearrange and complete the square in r:\n\n  (r−5/2)²+z² ≤(5/2)²−5 =5/4.\n\nThus in the rz–plane we have the filled disk\n\n  (r−5/2)²+z² ≤5/4,\n\nof radius a=√5/2 centered at (5/2,0).  Revolving this disk about the z–axis produces a solid torus with major radius R=5/2 and minor radius a=√5/2.  By Pappus’s centroid theorem, the volume is\n\n  V = (area of generating disk) × (distance traveled by its centroid)\n    = (πa²) × (2πR)\n    = (π·(5/4)) × (2π·(5/2))\n    =25π²/4.\n\nEquivalently, using the standard torus formula V=2π²Ra² gives the same result.",
      "_meta": {
        "core_steps": [
          "Convert to cylindrical coordinates so the inequality involves r and z only",
          "Take the square root and complete the square to obtain (r−R)^2+z^2≤a^2",
          "Interpret that inequality as a disc in the rz-plane that is revolved about the z-axis, forming a solid torus",
          "Use Pappus’s centroid theorem: volume = (area of the disc) × (distance traveled by its centroid)",
          "Insert the specific area and centroid path length to get the final volume"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Additive constant inside the squared expression (controls torus tube radius)",
            "original": "8"
          },
          "slot2": {
            "description": "Squared coefficient multiplying (x^2+y^2) on the right-hand side (controls torus major radius)",
            "original": "36"
          },
          "slot3": {
            "description": "Resulting major radius of the torus after completing the square (half the linear coefficient)",
            "original": "3"
          },
          "slot4": {
            "description": "Resulting minor (tube) radius of the torus after completing the square",
            "original": "1"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}