积极 AI Logo 积极 AI

收银机找零与多边形随机作图：条件概率与取整利息演示

收银机找零与多边形随机作图：条件概率与取整利息演示
几何画板及绘图内容加载中，请稍等……
2021-03-26 21:48
多边形微积分极限概率统计几何作图
本例通过“收银机找零”场景的随机几何构造，展示硬币/金额在不同条件下的取值与计数效果，并用图形联动参数变化。核心数学概念包括：条件概率（在给定金额/找零条件下的可行情况）、极限与利息计算（以金额规则随次数或步数变化的趋势为线索）、以及由多段线与圆等要素形成的几何求交（secant/截线思想在交点判定中体现）。作图要点：利用 GeoGebra 的 Point 创建关键点（如 pennycenter 等中心点），Polygon 构造多边形区域（对应纸面上的“账面/区域”），Shuffle 与 Random/RandomElement 生成随机选择或随机顺序，用 Intersect 计算随机配置下的交点，从而驱动后续的显示与计数；If 条件语句用于根据金额阈值/可行性筛选对象；Element 用于引用被随机抽取的对象元素。对象层面包含大量点与标记（A、B、C…与带下标的 I_4、Z_3、W_3 等），以及 pennycenter、pennysize 与分组对象（R_i、S_i 等）来表示“硬币尺寸/数量层”的几何刻画。适合高中到大学的数学建模与可视化教学，用于概率统计、几何作图与金融类“计息/找零规则”建模的综合演示。

GeoGebra 命令及参数

  1. Point(xAxis) → C_4
  2. pennycenter(R_{11}, pennysize) → T
  3. pennycenter(R_{13}, pennysize) → U
  4. pennycenter(R_{14}, pennysize) → V
  5. pennycenter(R_{15}, pennysize) → W
  6. pennycenter(R_{16}, pennysize) → Z
  7. pennycenter(R_{10}, pennysize) → A_1
  8. pennycenter(R_{17}, pennysize) → B_1
  9. pennycenter(R_{18}, pennysize) → C_1
  10. pennycenter(R_{19}, pennysize) → D_1
  11. pennycenter(R_1, pennysize) → E_1
  12. pennycenter(R_2, pennysize) → F_1
  13. pennycenter(R_3, pennysize) → G_1
  14. pennycenter(R_4, pennysize) → H_1
  15. pennycenter(R_5, pennysize) → I_1
  16. pennycenter(R_6, pennysize) → J_1
  17. pennycenter(R_7, pennysize) → K_1
  18. pennycenter(R_8, pennysize) → L_1
  19. pennycenter(R_9, pennysize) → M_1
  20. pennycenter(R_{12}, pennysize) → N_1
  21. pennycenter(O_1, nickelsize) → Q_1
  22. pennycenter(O_2, nickelsize) → T_1
  23. pennycenter(O_3, nickelsize) → U_1
  24. pennycenter(O_4, nickelsize) → V_1
  25. pennycenter(O_5, nickelsize) → W_1
  26. pennycenter(O_6, nickelsize) → Z_1
  27. pennycenter(O_7, nickelsize) → A_2
  28. pennycenter(O_8, nickelsize) → B_2
  29. pennycenter(O_9, nickelsize) → C_2
  30. pennycenter(O_{10}, nickelsize) → D_2
  31. pennycenter(O_{11}, nickelsize) → E_2
  32. pennycenter(O_{12}, nickelsize) → F_2
  33. pennycenter(O_{13}, nickelsize) → G_2
  34. pennycenter(O_{14}, nickelsize) → H_2
  35. pennycenter(O_{15}, nickelsize) → I_2
  36. pennycenter(O_{16}, nickelsize) → J_2
  37. pennycenter(O_{17}, nickelsize) → K_2
  38. pennycenter(O_{18}, nickelsize) → L_2
  39. pennycenter(O_{19}, nickelsize) → M_2
  40. pennycenter(O_{20}, nickelsize) → N_2
  41. Polygon(J_3, K_3, C_4, D_4, E_4, F_4, G_4, I_3, L_3, M_3, A, V_3, W_3, Z_3, A_4, B_4) → poly4, h, r, s, t, f_1, g_1, h_1, i_1, i, j, k, l, m, n, p, q
  42. pennycenter(E_6, dimesize) → D_6
  43. pennycenter(E_7, dimesize) → F_6
  44. pennycenter(E_8, dimesize) → G_6
  45. pennycenter(E_9, dimesize) → H_6
  46. pennycenter(E_{10}, dimesize) → I_6
  47. pennycenter(E_{11}, dimesize) → J_6
  48. pennycenter(E_{12}, dimesize) → K_6
  49. pennycenter(E_{13}, dimesize) → L_6
  50. pennycenter(E_{14}, dimesize) → M_6
  51. pennycenter(E_{15}, dimesize) → N_6
  52. pennycenter(E_{16}, dimesize) → Q_6
  53. pennycenter(E_{17}, dimesize) → T_6
  54. pennycenter(E_{18}, dimesize) → U_6
  55. pennycenter(E_{19}, dimesize) → V_6
  56. pennycenter(E_{21}, dimesize) → W_6
  57. pennycenter(E_{22}, dimesize) → Z_6
  58. pennycenter(E_{23}, dimesize) → A_7
  59. pennycenter(E_{24}, dimesize) → B_7
  60. pennycenter(E, dimesize) → D_7
  61. pennycenter(E_{20}, dimesize) → F_7
  62. pennycenter(G_7, quartersize) → I_7
  63. pennycenter(G_{10}, quartersize) → J_7
  64. pennycenter(G_9, quartersize) → K_7
  65. pennycenter(G_8, quartersize) → L_7
  66. pennycenter(G_{14}, quartersize) → M_7
  67. pennycenter(G_{13}, quartersize) → N_7
  68. pennycenter(G_{12}, quartersize) → Q_7
  69. pennycenter(G_{11}, quartersize) → T_7
  70. pennycenter(G_{18}, quartersize) → U_7
  71. pennycenter(G_{17}, quartersize) → V_7
  72. pennycenter(G_{16}, quartersize) → W_7
  73. pennycenter(G_{15}, quartersize) → Z_7
  74. pennycenter(G_{19}, quartersize) → A_8
  75. pennycenter(G_{20}, quartersize) → B_8
  76. pennycenter(G_{21}, quartersize) → D_8
  77. pennycenter(G_{22}, quartersize) → F_8
  78. pennycenter(G_{23}, quartersize) → I_8
  79. pennycenter(G_{24}, quartersize) → J_8
  80. pennycenter(G_{25}, quartersize) → K_8
  81. pennycenter(G_{26}, quartersize) → L_8
  82. Shuffle(Remove[Flatten[Sequence[Sequence[Point[{i, j}], i, -32, -22, 3], j, 8, 0, -3]], {(-32, 8), (-23, 2)}]) → leftbanklist, {(-29, 5), (-29, 2), (-32, 5), (-26, 8), (-23, 8), (-32, 2), (-26, 5), (-23, 5), (-29, 8), (-26, 2)}
  83. Polygon(C_4, B_4, A_4, Z_3, E_4, D_4) → poly1, c_4, b_4, a_4, z_3, e_4, d_4
  84. Polygon(B, C, D, F) → poly2, b_1, c_1, d_1, f_2
  85. Intersect(z_3, e_4) → J
  86. Polygon(J, F_4, W_3, Z_3) → poly3, j_2, f_4, w_3, z_1
  87. Polygon(F_4, G_4, V_3, W_3) → poly5, f_3, g_4, v_3, w_1
  88. Polygon(G_4, I_3, A, V_3) → poly6, g_2, i_3, a_1, v_1
  89. Polygon(H_4, I_4, N_3, U_3) → poly7, h_4, i_4, n_3, u_3
  90. Polygon(J_3, K_3, L_3, M_3) → poly8, j_3, k_3, l_3, m_3
  91. Polygon(K, L, M, N) → poly9, k_2, l_2, m_2, n_1
  92. Polygon(O, P, Q, Q_2) → poly10, o, p_1, q_1, q_2
  93. RandomElement({"gum", "yo-yo", "bottle of bubbles", "top", "candy", "lollipop", "pencil"}) → buying, "top"
  94. Element({"1¢", "10¢", "20¢", "50¢", "$1.00"}, u) → maxText
  95. Random(Element[{1, 10, 20, 50, 100}, u - 1], Element[{1, 10, 20, 50, 100}, u]) → cost
  96. If(cost ≟ 100, "$1.00", Text[cost] + "¢") → costText
  97. If(Rtotal ≟ 0, "0¢", If[Rtotal < 100, Rtotal + "¢", "$" + Rhundreds + "." + Rtens + Rones]) → text2
  98. If(cost ≟ Rtotal, "You bought it!", If[cost > Rtotal, "You haven't paid enough.", "You paid too much."]) → checkText

AI 绘图提示语参考

在平面上准备一组以硬币大小为尺度的“现金找零配置区域”。先放置若干以中心点为锚的圆/点位置：一组用于一枚便士大小的中心点（包含多个命名中心点，并形成一个较大的多边形边界），另一组用于五分钱大小的中心点，再在其周围再放置一组十分钱大小的中心点。随后以便士大小相关的若干中心点为顶点依次连成一个复杂的多边形轮廓；同时用这些顶点作为分区边界，使多边形覆盖到由不同中心点构成的区域。最后再结合更多便士相关中心点补足多边形的下界与左右扩展，并确保所有锚点与多边形顶点在同一平面坐标体系中。
以上提示语为根据原始绘图命令整理而来，仅供参考。