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Box Folding Problem (basic version)

Box Folding Problem (basic version)
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2013-09-10 01:07
FunctionSolid GeometryCalculusDerivativeOptimization
This example demonstrates the process of cutting and folding a rectangular piece of cardboard to form a box, focusing on determining the cut length to maximize the volume of the resulting box. Key mathematical concepts include volume calculation, derivative application, function optimization, and geometric transformations. GeoGebra commands used in construction are: Function, Segment, Rotate, Line, Intersect, Angle, Polygon, Derivative, Root, and Min. It is suitable for high school students to understand optimization problems in real-world applications using 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

AI Drawing Prompt Reference

Draw a rectangular piece of cardboard and fold it along the diagonal to form an open-top box. Start by defining the length and width of the rectangle, then cut small squares from each corner so that the sides can fold to form the box's base. Use rotations and lines to find the cutting length that maximizes the box's volume. Finally, apply functions and derivatives to determine this optimal solution.
The prompt above was compiled from the original drawing commands and is for reference only.