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Folding Box for Maximum Volume

Folding Box for Maximum Volume
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2020-08-02 15:09
CalculusOptimizationGeometric Construction
This GeoGebra example demonstrates the problem of folding a box to achieve the maximum volume. Key mathematical concepts include volume calculation and optimization. Using GeoGebra commands such as Function, Segment, Rotate, Line, Intersect, Angle, Polygon, Derivative, Root, Min, a dynamic box model is constructed to show how adjusting length and width maximizes the volume. Suitable for high school mathematics education and learning.

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 dynamic geometric figure illustrating the folding box maximum volume problem. Start with a rectangle representing the cardboard, with four vertices. From one corner, construct two segments along the edges to indicate the fold parts. Use rotation and parallel lines to determine intersection points of the folded faces, forming four side polygons. Beside the figure, plot a curve showing the box volume as a function of the fold height, and mark the extremum point on the curve, which corresponds to the maximum volume. Connect this extremum point to a relevant point in the figure with a segment to show the optimal fold height. The figure should include the rectangle, fold lines, side polygons, volume curve, and extremum marker.
The prompt above was compiled from the original drawing commands and is for reference only.