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Volume of Planter Box: Modeling with Functions, Derivatives, and Linear Regressi

Volume of Planter Box: Modeling with Functions, Derivatives, and Linear Regressi
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2016-09-09 15:59
FunctionSolid GeometryDerivativeOptimizationGraphing
This GeoGebra example demonstrates how to model the volume of a planter box using functions, derivatives, and linear regression. It visualizes the relationship between box dimensions and volume, and explores optimization through derivative-based extremum finding. Core mathematical concepts include volume calculation, function modeling, derivative for optimization, linear regression for data fitting, and scatter plot analysis. The construction uses commands such as Function, Segment, Rotate, Line, Intersect, Angle, Polygon, Derivative, Root, and Min, along with sliders to dynamically adjust parameters. It is suitable for high school or early college students learning applied calculus, data modeling, and optimization in real-world contexts, and can serve as a cross-disciplinary project in engineering design.

GeoGebra Commands & Parameters

  1. Function(((x * (length_1 - (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) → l_3
  12. Segment(C, D) → c
  13. Intersect(s1, c) → Dx
  14. Segment(Ax, Dx) → d1
  15. Intersect(d1, l_3) → D1
  16. Line(B, s) → q
  17. Line(Bx, q) → q1
  18. Intersect(q1, c) → Cx
  19. Segment(Bx, Cx) → b1
  20. Intersect(b1, l_3) → 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_2
  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
  37. Intersect(l_3, d1) → E
  38. Intersect(l_3, b1) → F
  39. Intersect(b1, a1) → G
  40. Intersect(d1, a1) → H
  41. Polygon(E, F, G, H) → poly5, e_2, f_2, g_2, h_2
  42. Intersect(l_3, d1) → I
  43. Intersect(l_3, b1) → J
  44. Segment(I, J) → len1
  45. Intersect(d1, a1) → K
  46. Segment(I, K) → j_1
  47. Intersect(l_3, d1) → N
  48. Intersect(l_3, b1) → O
  49. Segment(N, O) → m_1
  50. Intersect(k, l_2) → Q
  51. Intersect(c, c_1) → R
  52. Intersect(b, c) → T