MathPlotAI Logo MathPlotAI

Maximizing Trapezoid Area

Maximizing Trapezoid Area
Loading the math board and drawing, please wait…
2016-07-16 11:43
CalculusOptimizationGeometric Construction
In collections AREA Optimization: Minima and Maxima
This example demonstrates how to maximize the area of an isosceles trapezoid by dynamically adjusting its parameters using calculus concepts like optimization and derivatives. Key GeoGebra commands used include Circle, Intersect, Rotate, Angle, Segment, Mirror, CircleSector, Vector, Polygon, Dilate, and Translate to construct the geometric figure and compute its area. Suitable for high school and above students learning optimization problems and geometric transformations in calculus.

GeoGebra Commands & Parameters

  1. Circle(B, 1, xOyPlane) → c
  2. Intersect(h, c) → G, H
  3. Rotate(G, (-θ), B, xOyPlane) → G'
  4. Angle(G', B, G, xOyPlane) → β
  5. Segment(B, G') → f
  6. Segment(B, G) → g
  7. Mirror(G', l) → G''_1
  8. Segment(A, G''_1) → m
  9. Segment(A, B) → n
  10. Segment(G''_1, G') → i
  11. Intersect(h, i) → I
  12. Mirror(I, l) → I'
  13. Circle(B, 1.5, xOyPlane) → d
  14. Circle(A, 1.5, xOyPlane) → e
  15. Intersect(p, e) → J, L
  16. Intersect(h, d) → M, N
  17. Angle(I', A, G''_1, xOyPlane) → γ
  18. Circle(A, 0.2, xOyPlane) → k
  19. Circle(B, 0.2, xOyPlane) → q
  20. Intersect(f, q) → O,
  21. Intersect(h, q) → P, Q
  22. Intersect(p, k) → R, S
  23. Intersect(m, k) → T,
  24. CircleSector(B, O, P, xOyPlane) → r
  25. CircleSector(A, R, T, xOyPlane) → s
  26. Circle(A, 1, xOyPlane) → t
  27. Intersect(p, t) → K, U
  28. Mirror(K, l) → K'
  29. Vector(A, K) → u
  30. Vector(B, K') → v
  31. Angle(K', B, A, xOyPlane) → δ
  32. Angle(B, A, K, xOyPlane) → ε
  33. Segment(A, I') → j
  34. Segment(B, I) → b
  35. Polygon(A, B, G', G''_1) → poly1, a_1, b_1, g', g''_1
  36. Dilate(G''_1, If[0 < o ≤ 1, 1 - o, 0], A) → D
  37. Segment(G''_1, D) → f_1
  38. Rotate(G''_1, If[1 < o ≤ 2, (-(π / 2 + θ)), (-(π / 2 + θ))], A) → F
  39. Rotate(G''_1, If[1 < o ≤ 2, ((-((-o) - 1)) * (π / 2 + θ)), (-(π / 2 + θ))], A) → V
  40. Rotate(D, If[1 < o ≤ 2, ((-((-o) - 1)) * (π / 2 + θ)), (-(π / 2 + θ))], A) → D'
  41. Angle(B, A, G''_1, xOyPlane) → α
  42. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], B) → A'
  43. Vector(A, A') → w
  44. Translate(G''_1, w) → W
  45. Translate(D, w) → D'_1
  46. Translate(G''_1, w) → Z
  47. Rotate(G''_1, If[1 < o ≤ 2, ((-(o - 1)) * (π / 2 + θ)), (-(π / 2 + θ))], A) → A_1
  48. Rotate(D, If[1 < o ≤ 2, ((-(o - 1)) * (π / 2 + θ)), (-(π / 2 + θ))], A) → D'_2
  49. Segment(A_1, D'_2) → h_1
  50. Rotate(A_1, If[2 < o ≤ 3, ((-(o - 2)) * (π / 2 + θ)), (-(π / 2 + θ))], B) → B_1
  51. Rotate(D'_2, If[2 < o ≤ 3, ((-(o - 2)) * (π / 2 + θ)), (-(π / 2 + θ))], B) → C_1
  52. Segment(B_1, C_1) → g_1
  53. Area(poly1) → c_1