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Box Folding for Maximum Volume

Box Folding for Maximum Volume
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2016-11-01 06:49
CalculusOptimizationGeometric Construction
This example demonstrates how to fold a rectangular sheet to form a box with maximum volume. Key mathematical concepts include geometric mean and derivatives. GeoGebra commands such as Function, Segment, Rotate, Line, and Intersect are used for geometric construction. Suitable for high school mathematics education and learning optimization problems in calculus.

GeoGebra Commands & Parameters

  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