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Maximizing Box Volume

Maximizing Box Volume
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2015-02-26 01:49
Solid GeometryCalculusOptimizationGeometric Construction
This example demonstrates how to maximize the volume of a box formed from a sheet of cardboard by removing a square from each corner. It involves core math concepts like volume calculation, optimization problems, similar triangles, and derivatives. Constructed using GeoGebra commands such as PointList, Point, Polygon, and Intersect. Suitable for high school mathematics education to understand practical applications of calculus and geometric optimization.

GeoGebra Commands & Parameters

  1. PointList(Transpose[(PM * Transpose[FL])]) → FP
  2. PointList(Transpose[(PM * Transpose[ML])]) → MP
  3. Point(FP) → A
  4. Point(FP) → B
  5. Point(FP) → C
  6. Point(FP) → D
  7. Polygon(A, B, C, D) → poly1, a, b, c, d
  8. Intersect(b, a) → E
  9. Point(MP) → F
  10. Point(MP) → G
  11. Intersect(c, b) → H
  12. Polygon(E, F, G, H) → poly2, e, f, g, h_1
  13. Intersect(c, b) → I
  14. Intersect(d, c) → J
  15. Point(MP) → K
  16. Point(MP) → L
  17. Polygon(I, J, K, L) → poly3, i, j, k, l_1
  18. Intersect(d, a) → M
  19. Intersect(d, c) → N
  20. Point(MP) → O
  21. Point(MP) → P
  22. Polygon(N, M, O, P) → poly4, n, m, o, p
  23. Intersect(b, a) → Q
  24. Point(MP) → R
  25. Point(MP) → S
  26. Intersect(d, a) → T
  27. Polygon(Q, R, S, T) → poly5, q_1, r, s, t
  28. Intersect(r, q_1) → U
  29. Point(FP) → V
  30. Intersect(f, e) → W
  31. Intersect(b, a) → Z
  32. Polygon(U, V, W, Z) → poly6, u, v, w_1, z
  33. Intersect(g, f) → A_1
  34. Intersect(c, b) → B_1
  35. Intersect(l_1, k) → C_1
  36. Point(FP) → D_1
  37. Polygon(A_1, B_1, C_1, D_1) → poly7, a_1, b_1, c_1, d_1
  38. Intersect(s, r) → E_1
  39. Point(FP) → F_1
  40. Intersect(o, m) → G_1
  41. Intersect(d, a) → H_1
  42. Polygon(E_1, F_1, G_1, H_1) → poly8, e_1, f_1, g_1, h_2
  43. Intersect(j, p) → I_1
  44. Intersect(p, o) → J_1
  45. Point(FP) → K_1
  46. Intersect(k, j) → L_1
  47. Polygon(I_1, J_1, K_1, L_1) → poly9, i_1, j_1, k_1, l_2