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纸盒折叠求最大体积

纸盒折叠求最大体积
几何画板及绘图内容加载中，请稍等……
2016-11-01 06:49
微积分最值优化几何作图
这个范例演示了如何通过折叠矩形纸片形成一个具有最大体积的纸盒。核心数学概念包括几何均值和导数。通过使用GeoGebra命令如Function, Segment, Rotate, Line, Intersect等进行几何作图。适合高中数学教学和学习微积分中的最值优化问题。

GeoGebra 命令及参数

  1. Function(((x * (length - (2 * x))) * (width - (2 * x))), 0, Vright) → V
  2. Segment(B, C) → b
  3. Rotate(B, α, A) → B'
  4. Line(B', A) → s
  5. Line(Ax, s) → s1
  6. Segment(A, B) → a
  7. Line(C, a) → r
  8. Intersect(s, r) → D
  9. Line(Dy, r) → r1
  10. Intersect(r1, b) → Cy
  11. Segment(Dy, Cy) → c1
  12. Segment(C, D) → c
  13. Intersect(s1, c) → Dx
  14. Segment(Ax, Dx) → d1
  15. Intersect(d1, c1) → D1
  16. Line(B, s) → q
  17. Line(Bx, q) → q1
  18. Intersect(q1, c) → Cx
  19. Segment(Bx, Cx) → b1
  20. Intersect(b1, c1) → C1
  21. Line(A, B) → p
  22. Line(Ay, p) → p1
  23. Intersect(p1, b) → By
  24. Segment(Ay, By) → a1
  25. Intersect(d1, a1) → A1
  26. Intersect(b1, a1) → B1
  27. Angle(B, A, B') → β
  28. Segment(D, A) → d
  29. Polygon(A1, A2a, B2a, B1) → poly1, e, f, g, h_1
  30. Polygon(B1, B2b, C2b, C1) → poly2, i, j, k, l
  31. Polygon(C1, C2c, D2c, D1) → poly3, c_1, d_1, e_1, f_1
  32. Polygon(D1, D2d, A2d, A1) → poly4, m, n, a_1, b_1
  33. Derivative(V) → Vder
  34. Root(Vder) → tMax_1, tMax_2
  35. Min(x(tMax_1), x(tMax_2)) → tMax
  36. Segment(tMax_1, P) → g_1
  37. Intersect(c1, d1) → E
  38. Intersect(b1, c_1) → F
  39. Intersect(c1, b1) → G
  40. Intersect(b1, a1) → H
  41. Intersect(d1, a1) → I
  42. Polygon(E, G, H, I) → poly5, e_2, g_2, h_2, i_1
  43. Tangent(P, V) → j_1
  44. Point(j_1) → J
  45. Point(j_1) → K
  46. Segment(J, K) → k_1