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Volume Comparison of Sphere, Pyramid, and Cone

Volume Comparison of Sphere, Pyramid, and Cone
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2016-05-23 21:21
Solid GeometryGeometric Construction
This example demonstrates the volume changes of a sphere with a radius of 5, a square pyramid with a base of 10x10, and a cone with a base radius of 5 when water is added at the same rate. The core mathematical concepts involved are volume calculations in solid geometry. Using GeoGebra commands such as Surface, Intersect, If, and Distance, these geometric shapes can be constructed and their volume changes observed. Suitable for high school math education or students learning solid geometry.

GeoGebra Commands & Parameters

  1. Surface(t, u, abs(t + u) + abs(t - u), t, -5, 5, u, -5, 5) → c
  2. Surface(((0.5 * t) * cos(u)) + 12, ((0.5 * t) * sin(u)), t, u, 0, (2 * π), t, 0, 10) → b
  3. Intersect(sph, f, (2.553733711495868, 84.99999998631596)) → S
  4. Intersect(pyr, f, (6.341325703714092, 84.99999993280886)) → P
  5. Intersect(con, f, (6.87305943653345, 84.99999994614204)) → C
  6. If(0 ≤ y(A) ≤ (250 * π / 3), Distance[A, C], 10) → wc
  7. If(0 ≤ y(A) ≤ 1000 / 3, (0.5 * Distance[A, P]), 5) → wp
  8. Distance(A, S) → ws
  9. Surface(((0.5 * t) * cos(u)) + 12, ((0.5 * t) * sin(u)), t, u, 0, (2 * π), t, 0, wc) → e
  10. Surface(t, u, abs(t + u) + abs(t - u), t, (-wp), wp, u, (-wp), wp) → g
  11. Surface((sqrt((10 * u) - u^(2)) * cos(t)) - 10, (sqrt((10 * u) - u^(2)) * sin(t)), u, u, 0, ws, t, 0, (2 * π)) → h

AI Drawing Prompt Reference

Draw a sphere with a radius of 5, a square pyramid with a base of 10x10, and a cone with a base radius of 5, and observe the changes in their volumes as water is added at the same rate. Start by drawing the sphere, then the pyramid and cone, and finally adjust the surface display range dynamically based on the water level height.
The prompt above was compiled from the original drawing commands and is for reference only.