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Modifiable Solid with Equilateral Triangular Cross Sections

Modifiable Solid with Equilateral Triangular Cross Sections
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2019-04-18 20:36
FunctionSolid GeometryCalculusIntegral
This example demonstrates constructing a solid where the cross sections parallel to the y-axis are equilateral triangles, with side lengths equal to the difference between the upper and lower functions. Key mathematical concepts include equilateral triangles, functions, and integrals. GeoGebra commands used include Segment, Midpoint, Intersect, Line, Polygon, Textfield, CurveCartesian, Surface, and Integral. Suitable for high school and above students learning solid geometry and calculus.

GeoGebra Commands & Parameters

  1. Segment(H, I) → s_1
  2. Midpoint(s_1) → E
  3. Segment(J, K) → q_1
  4. Midpoint(q_1) → F
  5. Intersect(upper, h, (4.29, 7.239385)) → B
  6. Intersect(lower, h, (4.29, -4.3871887808018935)) → C
  7. Segment(C, B) → q
  8. Intersect(upper, q, (4.29, 7.239385)) → G
  9. Midpoint(C, G) → T
  10. Line(B, D) → i
  11. Segment(C, B) → p
  12. Polygon(C, G, U) → q1, f_1, g_1, n_1
  13. Polygon(K, J, W) → q2, l, m, n
  14. Polygon(Q, P, V) → q3, r, s, t
  15. Textfield(upper) → InputBox1
  16. Textfield(lower) → InputBox2
  17. Textfield(a) → InputBox3
  18. Textfield(b) → InputBox4
  19. Segment(K, W) → f
  20. Segment(W, J) → g
  21. Segment(Q, V) → j
  22. Segment(V, P) → k
  23. CurveCartesian(t, (0.5 * (upper(t) + lower(t))), ((sqrt(3) * 1 / 2) * (upper(t) - lower(t))), t, a, b) → c
  24. Surface(u, ((1 - v) * lower(u)) + ((0.5 * v) * (lower(u) + upper(u))), (sqrt(3) / 2 * (v * (upper(u) - lower(u)))), u, a, b, v, 0, 1) → o
  25. Surface(u, (v * upper(u)) + (((1 - v) * 0.5) * (lower(u) + upper(u))), (((sqrt(3) * 0.5) * (1 - v)) * (upper(u) - lower(u))), u, a, b, v, 0, 1) → u
  26. Segment(C, U) → i_1
  27. Segment(U, G) → j_1
  28. Segment(C, G) → k_1
  29. Integral((t_1(x) * t_1(x)), a, b) → a_1
  30. Integral((t_1(x) - h_1(x))^(2), a, b) → w

AI Drawing Prompt Reference

Create a solid where the cross sections parallel to the y-axis are equilateral triangles, with side lengths equal to the difference between the upper and lower functions. First, define the upper and lower functions, then construct equilateral triangular cross sections within the specified interval, and finally connect these cross sections to form the solid.
The prompt above was compiled from the original drawing commands and is for reference only.