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Classic Box of Maximum Volume Visualization

Classic Box of Maximum Volume Visualization
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2012-08-13 04:08
Solid GeometryOptimizationSlider Animation
In collections Area, perimeter and volume of shapes Area, Perimeter and Volume of shapes
This example demonstrates how to adjust the dimensions of a box by moving sliders g, h, and height to observe changes in volume. Key mathematical concepts include solid geometry and derivatives. Commands used include Intersect, Polygon, Mirror, If, Vector, Derivative, Midpoint, Max, and Name. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → D
  2. Polygon(D, A, C, B) → poly1, d, a, c, b
  3. Polygon(D, F, E, G) → poly2, d_1, f, e, g_1
  4. Mirror(D, x = Segment[D, A] / 2) → D'
  5. Mirror(F, x = Segment[D, A] / 2) → F'
  6. Mirror(E, x = Segment[D, A] / 2) → E'
  7. Mirror(G, x = Segment[D, A] / 2) → G'
  8. Polygon(D', F', E', G') → poly2', d', f', e', g'
  9. Mirror(D, y = Segment[D, B] / 2) → D'_1
  10. Mirror(A, y = Segment[D, B] / 2) → A'
  11. Mirror(C, y = Segment[D, B] / 2) → C'
  12. Mirror(B, y = Segment[D, B] / 2) → B'
  13. Mirror(D, y = Segment[D, B] / 2) → D'_2
  14. Mirror(F, y = Segment[D, B] / 2) → F'_1
  15. Mirror(E, y = Segment[D, B] / 2) → E'_1
  16. Mirror(G, y = Segment[D, B] / 2) → G'_1
  17. Polygon(D'_2, F'_1, E'_1, G'_1) → poly2'_1, d'_2, f'_1, e'_1, g'_1
  18. If(Segment[D, B] ≥ Segment[D, A], Segment[D, A] / 2, Segment[D, B] / 2) → maxval
  19. If(Segment[D, B] ≥ Segment[D, A], maxval = Segment[D, A] / 2, maxval = Segment[D, B] / 2) → i
  20. Mirror(D', y = Segment[A, C] / 2) → D''
  21. Mirror(F', y = Segment[A, C] / 2) → F''
  22. Mirror(E', y = Segment[A, C] / 2) → E''
  23. Mirror(G', y = Segment[A, C] / 2) → G''
  24. Polygon(D'', F'', E'', G'') → poly2'', d'', f'', e'', g''
  25. Polygon(E'_1, E'', E', E) → poly3, e'_2, e''_1, e'_3, e_1
  26. Vector(H, I) → u
  27. Derivative(V) → p
  28. Midpoint(D, F) → dfavg
  29. Max(V, 0, maxval) → J
  30. Max(V, 0, maxval) → maxofv
  31. Name(height) → A1