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Half-Life Dynamic Geometric Illustration

Half-Life Dynamic Geometric Illustration
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2017-01-01 10:57
FunctionCalculusTransformation
In collections Exponential & Logarithmic Functions Exponential & Logarithmic Functions
This GeoGebra example dynamically visualizes the half-life of a radioactive element, illustrating exponential decay through interactive geometric transformations. Users adjust initial quantity and decay coefficient via input boxes or sliders, observing real-time changes in the decay curve and associated geometric figures. Core mathematical concepts include exponential functions, half-life formulas, and the correspondence between geometric dilations and proportional decay rates. Construction leverages GeoGebra commands such as Textfield for parameter input, Dilate, Rotate, Translate, Vector, Ray, and Integral for computing accumulated decay or area under the curve, alongside Intersect, Segment, and Midpoint for precise point and segment management. The model is suitable for high school exponential function units, calculus applications of integrals in accumulation contexts, or physics-based radioactive decay modeling, serving both teacher-led demonstrations and student-driven inquiry.

GeoGebra Commands & Parameters

  1. Textfield(a_0) → InputBox1
  2. Textfield(k) → InputBox2
  3. Intersect(A, b, (475.00000000000006, 608.7077589414912)) → D
  4. Intersect(A, c, (950, 304.35387947074565)) → E
  5. Intersect(A, d, (1425, 152.17693973537283)) → F
  6. Intersect(A, e, (1900, 76.08846986768641)) → G
  7. Textfield(f) → InputBox3
  8. Intersect(b, h) → H
  9. Segment(H, D) → i
  10. Intersect(c, j) → I
  11. Intersect(d, l) → J
  12. Segment(I, E) → m
  13. Segment(J, F) → n
  14. Segment(D, I) → p
  15. Segment(E, J) → q
  16. Intersect(h, s) → K
  17. Intersect(r, t) → L
  18. Segment(K, C) → f_1
  19. Segment(C, L) → g_1
  20. Midpoint(M, B) → N
  21. Dilate(M, If[0 < o ≤ 1, 1 - o, 0], N) → M'
  22. Rotate(M, If[1 < o ≤ 2, ((o - 1) * π), π], N) → M'_1
  23. Rotate(M', If[1 < o ≤ 2, ((o - 1) * π), π], N) → M''
  24. Dilate(N, If[2 < o ≤ 3, 3 - o, 0], D) → N'
  25. Segment(N, N') → j_1
  26. Midpoint(D, O) → R
  27. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], N) → B'
  28. Segment(B, B') → h_1
  29. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], N) → B'_1
  30. Rotate(B', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], N) → B''
  31. Segment(B'_1, B'') → i_1
  32. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], R) → D'
  33. Segment(D, D') → k_1
  34. Dilate(R, If[4 < o ≤ 5, 5 - o, 0], O) → R'
  35. Segment(R, R') → l_1
  36. Rotate(D, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], R) → D'_1
  37. Rotate(D', If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], R) → D''
  38. Segment(D'_1, D'') → m_1
  39. Rotate(N, If[6 < o ≤ 7, ((-(o - 6)) * π), (-π)], D) → N'_1
  40. Rotate(N', If[6 < o ≤ 7, ((-(o - 6)) * π), (-π)], D) → N''
  41. Segment(N'_1, N'') → n_1
  42. Dilate(D, If[7 < o ≤ 8, 8 - o, 0], R) → D'_2
  43. Vector(D, D'_2) → u
  44. Translate(N'_1, u) → S
  45. Translate(N'', u) → N'''
  46. Segment(S, N''') → p_1
  47. Midpoint(P, E) → T
  48. Dilate(E, If[8 < o ≤ 9, 9 - o, 0], T) → E'
  49. Segment(E, E') → q_1
  50. Dilate(T, If[9 < o ≤ 10, 10 - o, 0], P) → T'
  51. Segment(T, T') → r_1
  52. Rotate(E, If[10 < o ≤ 11, ((-(o - 10)) * π), (-π)], T) → E'_1
  53. Rotate(E', If[10 < o ≤ 11, ((-(o - 10)) * π), (-π)], T) → E''
  54. Segment(E'_1, E'') → s_1
  55. Rotate(S, If[11 < o ≤ 12, ((-(o - 11)) * π), (-π)], E) → S'
  56. Rotate(N''', If[11 < o ≤ 12, ((-(o - 11)) * π), (-π)], E) → U
  57. Segment(S', U) → t_1
  58. Dilate(E, If[12 < o ≤ 13, 13 - o, 0], T) → E'_2
  59. Vector(E, E'_2) → v
  60. Translate(S', v) → S''
  61. Translate(U, v) → U'
  62. Segment(S'', U') → a_1
  63. Midpoint(F, Q) → V
  64. Dilate(F, If[13 < o ≤ 14, 14 - o, 0], V) → F'
  65. Segment(F, F') → b_1
  66. Dilate(V, If[14 < o ≤ 15, 15 - o, 0], Q) → V'
  67. Segment(V, V') → c_1
  68. Rotate(F, If[15 < o ≤ 16, ((-(o - 15)) * π), (-π)], V) → F'_1
  69. Rotate(F', If[15 < o ≤ 16, ((-(o - 15)) * π), (-π)], V) → F''
  70. Segment(F'_1, F'') → d_1
  71. Rotate(S'', If[16 < o ≤ 17, ((-(o - 16)) * π), (-π)], F) → S'''
  72. Rotate(U', If[16 < o ≤ 17, ((-(o - 16)) * π), (-π)], F) → U''
  73. Segment(S''', U'') → e_1
  74. Dilate(F, If[17 < o ≤ 18, 18 - o, 0], V) → F'_2
  75. Vector(F, F'_2) → w
  76. Translate(S''', w) → W
  77. Translate(U'', w) → U'''
  78. Segment(W, U''') → f_2
  79. Translate(U'', w) → U'''_1
  80. Ray((0, 0), (1, 0)) → g_2
  81. Integral(A, 0, (50 * k)) → o_1