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Water in the Glass: Rotation and Volume

Water in the Glass: Rotation and Volume
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2016-01-01 21:37
Solid GeometryCalculus
This example illustrates the rotation and volume changes of water in a glass. Key mathematical concepts include the calculation of the volume of a solid of revolution in solid geometry. Using commands such as rotation, intersect path, circle, and line, it demonstrates the volume changes of water at different rotation angles. Suitable for high school mathematics education, particularly in solid geometry and calculus.

GeoGebra Commands & Parameters

  1. RotateText("uStas\;\&\,Т.С.", 90°) → надпись01
  2. Rotate(Circle[Q, m, zAxis], γ, Line[A, yAxis]) → ω
  3. Cylinder(ω, (k * m)) → Ω, ω', a
  4. IntersectPath(ζ, Ω) → φ
  5. Circle(O_1, m, zAxis) → ω_1
  6. Cylinder(ω_1, (k * m)) → Ω_1, ω'_1, a_1
  7. Line(A + (0, 0, (0.5 * p)), xAxis) → d
  8. Parabola(F, d) → c
  9. If(h_1 > (k * m), acos(q - 1)) → β
  10. If(h_1 ≤ (k * m), (m * t), (((0.5 * k) * m) * (π + β + ((1 - q) * sqrt((q * (2 - q))))) / π)) → h_2
  11. Cylinder(ω_1, h_2) → Ω_2, ω'_2, a_2
  12. Intersect(Line[O_1 + (0, 0, h_2), xAxis], c, 1) → C
  13. CurveCartesian(x(A) + j, 0, z(A) - (0.5 * j^(2) / p), j, 0, jj) → θ
  14. Intersect(ζ, ω) → U_1, U_2
  15. Arc(ω, U_2, U_1) → ωω
  16. Arc(φ, U_2, U_1) → φ'