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恒定速率填充容器

恒定速率填充容器
几何画板及绘图内容加载中，请稍等……
2017-04-03 17:49
函数立体几何几何作图函数图像
①本范例演示了在恒定速率下填充容器的过程，通过动态图形展示液面上升的轨迹；②涉及的核心数学概念包括函数图像、逻辑推理、几何图形绘制、除法运算、中值计算以及棱柱体积计算；③作图要点提炼自构造摘要，使用了PointIn、Polygon、Point、Mirror、If、Arc、Join、Min、PolyLine、Intersect、Segment、RotateText等GeoGebra命令和对象，实现了动态液面和容器的精确绘制；④适合高中数学教学，用于讲解函数图像、几何体积及动态几何作图。

GeoGebra 命令及参数

  1. PointIn((x ≤ 2.5) ∧ (x ≥ 0.01) ∧ (y ≥ 0.01) ∧ (y ≤ 5)) → A2
  2. Polygon(Join[{Reverse[A1:A5], B1:B5}]) → poly5'
  3. Polygon(Join[{Reverse[A1:A3], B1:B3}]) → poly3'
  4. Point(Segment[(0, 0), (3, 0)]) → A1
  5. Mirror(A1, x = 3) → B1
  6. PointIn((x ≤ 2.49) ∧ (x ≥ 0) ∧ (y ≥ y(A2) + 0.01) ∧ (y ≤ 5.01)) → A3
  7. PointIn((x ≤ 2.5) ∧ (x ≥ 0.01) ∧ (y ≥ y(A3) + 0.01) ∧ (y ≤ 5.02)) → A4
  8. PointIn((x ≤ 2.49) ∧ (x ≥ 0) ∧ (y ≥ y(A4) + 0.01) ∧ (y ≤ 5.03)) → A5
  9. Mirror(A5, x = 3) → B5
  10. Mirror(A4, x = 3) → B4
  11. Mirror(A3, x = 3) → B3
  12. Mirror(A2, x = 3) → B2
  13. Mirror(A6, x = 3) → B6
  14. Polygon(A6, A5, B5, B6, A4, A3, A2, A1, B1, B2, B3, B4) → poly6, a6, b4_1, b5, b6, a5, a4_1, a3_1, a2_2, a1_3, b1_3, b2_2, b3_1
  15. Polygon(Join[{Reverse[A1:A2], B1:B2}]) → poly2'
  16. Polygon(Join[{Reverse[A1:A4], B1:B4}]) → poly4'
  17. If(t < C1, 0, 1) → h
  18. Arc(Circle[A7, A8, A9], A7, A9) → c
  19. Mirror(A7, x = 3) → B7
  20. Mirror(A7, x = 3) → A7'
  21. Mirror(A9, x = 3) → A9'
  22. Arc(Mirror[Circle[A7, A8, A9], x = 3], A7', A9') → c'
  23. Join({Reverse[A7:A1], B1:B7}) → list1
  24. Min(5.04, If[t < C1, (-3 + x(A1) + sqrt((3 - x(A1))^(2) + ((t * poly6) * ((-x(A2)) + x(A1)) / (y(A2) - y(A1))))) / (((-x(A2)) + x(A1)) / (y(A2) - y(A1))), If[t < C2, y(A2) + (-3 + x(A2) + sqrt((3 - x(A2))^(2) + (((t - C1) * poly6) * ((-x(A3)) + x(A2)) / (y(A3) - y(A2))))) / (((-x(A3)) + x(A2)) / (y(A3) - y(A2))), If[t < C3, y(A3) + (-3 + x(A3) + sqrt((3 - x(A3))^(2) + (((t - C2) * poly6) * ((-x(A4)) + x(A3)) / (y(A4) - y(A3))))) / (((-x(A4)) + x(A3)) / (y(A4) - y(A3))), If[t < C4, y(A4) + (-3 + x(A4) + sqrt((3 - x(A4))^(2) + (((t - C3) * poly6) * ((-x(A5)) + x(A4)) / (y(A5) - y(A4))))) / (((-x(A5)) + x(A4)) / (y(A5) - y(A4))), y(A5) + (-3 + x(A5) + sqrt((3 - x(A5))^(2) + (((t - C4) * poly6) * ((-x(A6)) + x(A5)) / (y(A6) - y(A5))))) / (((-x(A6)) + x(A5)) / (y(A6) - y(A5)))]]]]) → a
  25. PolyLine(list1) → g
  26. Intersect(f, g) → A, B
  27. Polygon(A, A1, B1, B) → poly1, a_1, a1, b1, b
  28. Polygon(A, A2, A1, B1, B2, B) → poly2, a_2, a2, a1_1, b1_1, b2, b_1
  29. Polygon(A, A3, A2, A1, B1, B2, B3, B) → poly3, d, e, f_1, g_1, h_1, i_1, j_1, k_1
  30. Polygon(A, A4, A3, A2, A1, B1, B2, B3, B4, B) → poly4, a_3, b_2, a4, a3, a2_1, a1_2, b1_2, b2_1, b3, b4
  31. Segment(A, B) → l_1
  32. Polygon(A, A5, B5, B, A4, A3, A2, A1, B1, B2, B3, B4) → poly5, i, s, m_1, n_1, j, k, l, m, n, p, q, r
  33. RotateText(text1, 90°) → text3

AI 绘图提示语参考

画一个容器逐渐被水填满的动态过程，展示水面上升的函数图像。容器为对称的圆柱体，底部从左到右依次有五个点，水面从左到右逐渐上升，用不同颜色的多边形表示水面和容器，水面用多段弧线连接，水面高度随时间变化，用函数图像表示，容器和水面之间用线段连接，整体动态展示水面的上升过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。