{
  "index": "1993-A-1",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "in the first quadrant as in the figure. Find $c$ so that the areas of the\ntwo shaded regions are equal. [Figure not included. The first region is\nbounded by the $y$-axis, the line $y=c$ and the curve; the other lies\nunder the curve and above the line $y=c$ between their two points of\nintersection.]",
  "solution": "Solution. Let \\( (b, c) \\) denote the rightmost intersection point. (See Figure 23.) We wish to find \\( c \\) such that\n\\[\n\\int_{0}^{b}\\left(\\left(2 x-3 x^{3}\\right)-c\\right) d x=0 .\n\\]\n\nThis leads to \\( b^{2}-(3 / 4) b^{4}-b c=0 \\). After substituting \\( c=2 b-3 b^{3} \\) and solving, we find a unique positive solution, namely \\( b=2 / 3 \\). Thus \\( c=4 / 9 \\). To validate the solution, we check that \\( (2 / 3,4 / 9) \\) is rightmost among the intersection points of \\( y=4 / 9 \\) and \\( y=2 x-3 x^{3} \\) : the zeros of \\( 2 x-3 x^{3}-4 / 9=(2 / 3-x)\\left(3 x^{2}+2 x-2 / 3\\right) \\) other than \\( 2 / 3 \\) are \\( (-1 \\pm \\sqrt{3}) / 3 \\), which are less than \\( 2 / 3 \\).",
  "vars": [
    "x",
    "y"
  ],
  "params": [
    "b",
    "c"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "alphaxvariable",
        "y": "alphayvariable",
        "b": "betaparam",
        "c": "charlieparam"
      },
      "question": "in the first quadrant as in the figure. Find $charlieparam$ so that the areas of the\ntwo shaded regions are equal. [Figure not included. The first region is\nbounded by the $alphayvariable$-axis, the line $alphayvariable=charlieparam$ and the curve; the other lies\nunder the curve and above the line $alphayvariable=charlieparam$ between their two points of\nintersection.]",
      "solution": "Solution. Let \\( (\\text{betaparam}, \\text{charlieparam}) \\) denote the rightmost intersection point. (See Figure 23.) We wish to find \\( \\text{charlieparam} \\) such that\n\\[\n\\int_{0}^{\\text{betaparam}}\\left(\\left(2 \\text{alphaxvariable}-3 \\text{alphaxvariable}^{3}\\right)-\\text{charlieparam}\\right) d \\text{alphaxvariable}=0 .\n\\]\n\nThis leads to \\( \\text{betaparam}^{2}-(3 / 4) \\text{betaparam}^{4}-\\text{betaparam}\\,\\text{charlieparam}=0 \\). After substituting \\( \\text{charlieparam}=2 \\text{betaparam}-3 \\text{betaparam}^{3} \\) and solving, we find a unique positive solution, namely \\( \\text{betaparam}=2 / 3 \\). Thus \\( \\text{charlieparam}=4 / 9 \\). To validate the solution, we check that \\( (2 / 3,4 / 9) \\) is rightmost among the intersection points of \\( \\text{alphayvariable}=4 / 9 \\) and \\( \\text{alphayvariable}=2 \\text{alphaxvariable}-3 \\text{alphaxvariable}^{3} \\) : the zeros of \\( 2 \\text{alphaxvariable}-3 \\text{alphaxvariable}^{3}-4 / 9=(2 / 3-\\text{alphaxvariable})\\left(3 \\text{alphaxvariable}^{2}+2 \\text{alphaxvariable}-2 / 3\\right) \\) other than \\( 2 / 3 \\) are \\( (-1 \\pm \\sqrt{3}) / 3 \\), which are less than \\( 2 / 3 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "waterfall",
        "y": "pineapple",
        "b": "dinosaur",
        "c": "notebook"
      },
      "question": "in the first quadrant as in the figure. Find $notebook$ so that the areas of the two shaded regions are equal. [Figure not included. The first region is bounded by the $pineapple$-axis, the line $pineapple=notebook$ and the curve; the other lies under the curve and above the line $pineapple=notebook$ between their two points of intersection.]",
      "solution": "Solution. Let \\( (dinosaur, notebook) \\) denote the rightmost intersection point. (See Figure 23.) We wish to find \\( notebook \\) such that\n\\[\n\\int_{0}^{dinosaur}\\left(\\left(2 waterfall-3 waterfall^{3}\\right)-notebook\\right) d waterfall=0 .\n\\]\n\nThis leads to \\( dinosaur^{2}-(3 / 4) dinosaur^{4}-dinosaur\\,notebook=0 \\). After substituting \\( notebook=2 dinosaur-3 dinosaur^{3} \\) and solving, we find a unique positive solution, namely \\( dinosaur=2 / 3 \\). Thus \\( notebook=4 / 9 \\). To validate the solution, we check that \\( (2 / 3,4 / 9) \\) is rightmost among the intersection points of \\( pineapple=4 / 9 \\) and \\( pineapple=2 waterfall-3 waterfall^{3} \\) : the zeros of \\( 2 waterfall-3 waterfall^{3}-4 / 9=(2 / 3-waterfall)\\left(3 waterfall^{2}+2 waterfall-2 / 3\\right) \\) other than \\( 2 / 3 \\) are \\( (-1 \\pm \\sqrt{3}) / 3 \\), which are less than \\( 2 / 3 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "knownvalue",
        "y": "horizontalaxis",
        "b": "leftmostpoint",
        "c": "variablevalue"
      },
      "question": "in the first quadrant as in the figure. Find $variablevalue$ so that the areas of the\ntwo shaded regions are equal. [Figure not included. The first region is\nbounded by the $horizontalaxis$-axis, the line $horizontalaxis=variablevalue$ and the curve; the other lies\nunder the curve and above the line $horizontalaxis=variablevalue$ between their two points of\nintersection.]",
      "solution": "Solution. Let \\( (leftmostpoint, variablevalue) \\) denote the rightmost intersection point. (See Figure 23.) We wish to find \\( variablevalue \\) such that\n\\[\n\\int_{0}^{leftmostpoint}\\left(\\left(2 knownvalue-3 knownvalue^{3}\\right)-variablevalue\\right) d knownvalue=0 .\n\\]\n\nThis leads to \\( leftmostpoint^{2}-(3 / 4) leftmostpoint^{4}-leftmostpoint variablevalue=0 \\). After substituting \\( variablevalue=2 leftmostpoint-3 leftmostpoint^{3} \\) and solving, we find a unique positive solution, namely \\( leftmostpoint=2 / 3 \\). Thus \\( variablevalue=4 / 9 \\). To validate the solution, we check that \\( (2 / 3,4 / 9) \\) is rightmost among the intersection points of \\( horizontalaxis=4 / 9 \\) and \\( horizontalaxis=2 knownvalue-3 knownvalue^{3} \\) : the zeros of \\( 2 knownvalue-3 knownvalue^{3}-4 / 9=(2 / 3-knownvalue)\\left(3 knownvalue^{2}+2 knownvalue-2 / 3\\right) \\) other than \\( 2 / 3 \\) are \\( (-1 \\pm \\sqrt{3}) / 3 \\), which are less than \\( 2 / 3 \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "b": "mncfleor",
        "c": "tpkasvud"
      },
      "question": "in the first quadrant as in the figure. Find $tpkasvud$ so that the areas of the\ntwo shaded regions are equal. [Figure not included. The first region is\nbounded by the $hjgrksla$-axis, the line $hjgrksla=tpkasvud$ and the curve; the other lies\nunder the curve and above the line $hjgrksla=tpkasvud$ between their two points of\nintersection.]",
      "solution": "Solution. Let \\( (mncfleor, tpkasvud) \\) denote the rightmost intersection point. (See Figure 23.) We wish to find \\( tpkasvud \\) such that\n\\[\n\\int_{0}^{mncfleor}\\left(\\left(2 qzxwvtnp-3 qzxwvtnp^{3}\\right)-tpkasvud\\right) d qzxwvtnp=0 .\n\\]\n\nThis leads to \\( mncfleor^{2}-(3 / 4) mncfleor^{4}-mncfleor tpkasvud=0 \\). After substituting \\( tpkasvud=2 mncfleor-3 mncfleor^{3} \\) and solving, we find a unique positive solution, namely \\( mncfleor=2 / 3 \\). Thus \\( tpkasvud=4 / 9 \\). To validate the solution, we check that \\( (2 / 3,4 / 9) \\) is rightmost among the intersection points of \\( hjgrksla=4 / 9 \\) and \\( hjgrksla=2 qzxwvtnp-3 qzxwvtnp^{3} \\) : the zeros of \\( 2 qzxwvtnp-3 qzxwvtnp^{3}-4 / 9=(2 / 3-qzxwvtnp)\\left(3 qzxwvtnp^{2}+2 qzxwvtnp-2 / 3\\right) \\) other than \\( 2 / 3 \\) are \\( (-1 \\pm \\sqrt{3}) / 3 \\), which are less than \\( 2 / 3 \\)."
    },
    "kernel_variant": {
      "question": "In the first quadrant consider the cubic curve\n\\[\n\\gamma:\\; y = 9x-2x^{3}.\n\\]\nLet the vertical line \\(x=1\\) and an as–yet–unknown horizontal line \\(y=c\\) cut the curve as shown.  Denote by \\(b>1\\) the abscissa of the right–hand intersection of the horizontal line with \\(\\gamma\\).\nTwo planar regions are shaded:\n  (i) the set of points lying between the curve, the line \\(x=1\\) and the line \\(y=c\\);  \n  (ii) the set of points that lie under the curve and above the line \\(y=c\\) between the two intersection points of \\(y=c\\) with the curve.\nFind the number \\(c>0\\) for which the two shaded regions have equal area.",
      "solution": "Corrected Solution.  Let f(x)=9x−2x³.  We seek c>0 and abscissae 1<a<b in the first quadrant with f(a)=f(b)=c such that\n\nRegion I: the set of points between the vertical line x=1, the curve y=f(x), and the horizontal line y=c, and\nRegion II: the set of points between the curve y=f(x) and the line y=c for x running from a to b,\nhave equal area.\n\n1.  Area of Region I.  On [1,a] one has f(x)<c, so the shaded vertical strip has area\n   S₁ = ∫₁ᵃ (c − f(x)) dx\n   = ∫₁ᵃ (c − (9x−2x³)) dx.\n\n2.  Area of Region II.  On [a,b] one has f(x)>c, so\n   S₂ = ∫ₐᵇ (f(x) − c) dx\n   = ∫ₐᵇ ((9x−2x³) − c) dx.\n\n3.  Equating S₁=S₂ gives\n   ∫₁ᵃ (c−f(x)) dx = ∫ₐᵇ (f(x)−c) dx\n   ⇒ ∫₁ᵇ (f(x)−c) dx =0.\n\n4.  But f(b)=c, so c = 9b − 2b³.  Also\n   ∫₁ᵇ f(x) dx = ∫₁ᵇ (9x−2x³) dx\n    = [ (9/2)x² − (1/2)x⁴ ]₁ᵇ\n    = (9/2 b² − 1/2 b⁴) − (9/2 − 1/2)\n    = 9/2 b² − 1/2 b⁴ − 4.\n\n   Hence ∫₁ᵇ(f−c)dx = (9/2 b² − 1/2 b⁴ − 4) − c(b−1) = 0\n   ⇒ multiply by 2 and substitute c=9b−2b³:\n     9b² − b⁴ − 8 − 2(9b−2b³)(b−1) = 0\n   ⇒ 3b⁴ −4b³ −9b² +18b −8 = 0.\n   Factor: (b−1)²(3b²+2b−8)=0.  Discard b=1 double‐root; solve 3b²+2b−8=0:\n     b = (−2 ± √100)/6 ⇒ b = 4/3 (positive).\n\n5.  Finally c = f(b) = 9·(4/3) − 2·(4/3)³ = 12 − 128/27 = 196/27.\n\nAnswer:  c = 196/27.\n\nRemark.  One checks that f′(x)=9−6x² has its maximum at x=√(3/2)≈1.225<4/3, so the intersection at x=4/3 is indeed the unique right‐most one, and the one at x=a∈(1,√(3/2)).",
      "_meta": {
        "core_steps": [
          "Express “equality of shaded areas” as ∫(curve − c) dx = 0 between the two x-intersections.",
          "Call the right intersection abscissa b and use the intersection condition c = f(b).",
          "Insert c into the area equation to get a single algebraic equation in b and solve for the positive root.",
          "Compute c from c = f(b).",
          "Verify that this (b, c) is indeed the rightmost intersection of the line and the curve."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Linear coefficient of the cubic curve y = αx − βx³",
            "original": "2"
          },
          "slot2": {
            "description": "Magnitude of the (negative) cubic coefficient of the curve",
            "original": "3"
          },
          "slot3": {
            "description": "Vertical line that serves as the left boundary of the region",
            "original": "x = 0 (the y-axis)"
          },
          "slot4": {
            "description": "Positive abscissa of the rightmost intersection point",
            "original": "b = 2/3"
          },
          "slot5": {
            "description": "Height of the horizontal line whose position is sought",
            "original": "c = 4/9"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}