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Filling a Cone: dV/dt vs dh/dt Constant

Filling a Cone: dV/dt vs dh/dt Constant
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2017-10-15 08:36
Solid GeometryCalculusDerivative
This example demonstrates the comparison between filling a right circular cone with liquid at a constant volume rate (dV/dt) and a constant height rate (dh/dt). It involves core calculus concepts like related rates, the cone volume formula, derivatives, and linear functions. Constructed using GeoGebra commands such as Dilate, Line, Intersect, Cone, Textfield, and Height, it is suitable for high school and above students learning about applications of derivatives in related rates problems.

GeoGebra Commands & Parameters

  1. Dilate(A, If[0 < t ≤ 1, 1 - t, 0], B) → A'
  2. Line(A, C) → f
  3. Intersect(a, f) → R
  4. Cone(A', A, Distance[A', R]) → b, c, d
  5. Dilate(B, If[1 < t ≤ 2, 2 - t, 0], D) → B'
  6. Intersect(e, f) → S
  7. Cone(B', A, Distance[B', S]) → g, h, i
  8. Dilate(D, If[2 < t ≤ 3, 3 - t, 0], F) → D'
  9. Intersect(j, f) → T
  10. Cone(D', A, Distance[D', T]) → l, q, m
  11. Dilate(F, If[3 < t ≤ 4, 4 - t, 0], H) → F'
  12. Intersect(n, f) → U
  13. Cone(F', A, Distance[F', U]) → o, r, s
  14. Dilate(H, If[4 < t ≤ 5, 5 - t, 0], J) → H'
  15. Intersect(u, f) → V
  16. Cone(H', A, Distance[H', V]) → v, c_1, w
  17. Dilate(J, If[5 < t ≤ 6, 6 - t, 0], L) → J'
  18. Intersect(a_1, f) → W
  19. Cone(J', A, Distance[J', W]) → b_1, d_1, e_1
  20. Dilate(L, If[6 < t ≤ 7, 7 - t, 0], N) → L'
  21. Intersect(f, f_1) → Z
  22. Cone(L', A, Distance[L', Z]) → g_1, h_1, i_1
  23. Dilate(N, If[7 < t ≤ 8, 8 - t, 0], P) → N'
  24. Intersect(j_1, f) → A_1
  25. Cone(N', A, Distance[N', A_1]) → k_1, p_1, l_1
  26. Textfield(k) → InputBox1
  27. Textfield(p) → InputBox2
  28. Cone(P, A, Radius[p_1]) → o_1, q_1, r_1
  29. Textfield(t) → InputBox3
  30. Height(o_1) → b_2
  31. Textfield(c_2) → InputBox4
  32. Cone((0, 0, c_2), A, c_2 / p) → d_2, e_2, f_2