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Volume Optimization of an Open Box from a Square

Volume Optimization of an Open Box from a Square
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2015-06-30 18:46
Solid GeometryOptimizationGeometric Construction
This example demonstrates the process of cutting squares from each corner of a 12cm x 12cm square piece of card and folding it into an open box, exploring volume optimization. Key math concepts include volume calculation, geometric transformation, and optimization. GeoGebra commands used are Polygon, Intersect, Line, Segment, Prism, Net, Volume, Textfield, and If. Suitable for high school geometry and calculus instruction on real-world optimization problems.

GeoGebra Commands & Parameters

  1. Polygon((0, 0), (k, 0), 4) → poly1
  2. Intersect(xAxis, yAxis) → B
  3. Line(E, G) → b
  4. Line(F, H) → c
  5. Line(E, F) → d
  6. Line(G, H) → e
  7. Intersect(d, poly1) → I, J
  8. Intersect(e, poly1) → K, L
  9. Intersect(b, poly1) → M, N
  10. Intersect(c, poly1) → O, P
  11. Segment(K, L) → g
  12. Segment(M, N) → h
  13. Segment(O, P) → i
  14. Polygon(I, E, G, K) → poly2, i_1, e_1, g_1, k_3
  15. Polygon(E, F, H, G) → poly3, e_2, f_1, h_1, g_2
  16. Polygon(M, O, F, E) → poly4, m, o, f_2, e_3
  17. Polygon(F, J, L, H) → poly5, f_3, j, l_3, h_2
  18. Polygon(G, H, P, N) → poly6, g_3, h_3, p, n
  19. Segment(A, C) → q
  20. Segment(A, B) → r
  21. Segment(C, D) → s
  22. Segment(B, D) → t
  23. Segment(I, E) → a_1
  24. Segment(K, G) → b_1
  25. Segment(G, N) → c_1
  26. Segment(H, P) → d_1
  27. Segment(H, L) → j_1
  28. Segment(F, J) → k_1
  29. Segment(F, O) → l_1
  30. Segment(M, E) → m_1
  31. Polygon(A, I, E, M) → poly7, a_2, i_2, e_4, m_2
  32. Polygon(K, G, N, C) → poly8, k_2, g_4, n_1, c_2
  33. Polygon(H, L, D, P) → poly9, h_4, l_2, d_2, p_1
  34. Polygon(O, F, J, B) → poly10, o_1, f_4, j_2, b_2
  35. Polygon(E, F, H, G) → poly11, q_1, r_1, s_1, t_1
  36. Prism(poly11, a) → u, Q, edgeBQ, edgeAR, edgeCS, edgeDT, edgeQR, edgeRS, edgeST, edgeQT, R, S, T, faceABQR, faceACSR, faceCDTS, faceBDTQ, faceQRST
  37. Net(u, fold) → v, U, F_1, G_1, H_1, I_1, J_1, faceUVWZ, face1, face3, face4, face2, V, face5, edgeUV, edgeVW, edgeWZ, edgeUZ, edge1, edge9, edge3, edge4, edge12, W, edge5, edge6, edge13, edge7, edge8, edge14, edge2, edge11, edge15, edge10, Z, A_1, B_1, C_1, D_1, E_1
  38. Segment(O, B) → n_2
  39. Segment(B, J) → p_2
  40. Volume(u) → w
  41. Segment(E, F) → q_2
  42. Textfield(f) → inputBox1
  43. If(k ≟ 12, Text[k, S_1], Text["k", S_1]) → text5
  44. If(k ≟ 12, Text[k, T_1], Text["k", T_1]) → text6

AI Drawing Prompt Reference

Start with a 12cm by 12cm square card. Cut squares of side length k from each corner. Fold the flaps to form an open box. Calculate the volume of the box and find the value of k that maximizes it. Ensure all steps follow geometric transformation and optimization principles.
The prompt above was compiled from the original drawing commands and is for reference only.