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Cylinder Volume with Fixed Surface Area

Cylinder Volume with Fixed Surface Area
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2015-05-13 09:42
Solid GeometryCalculusOptimizationGeometric Construction
This example demonstrates the relationship between the volume of a cylinder and its dimensions (radius and height) when the surface area is fixed. Key math concepts include cylinder surface area, volume formula, and optimization. GeoGebra commands/objects like r, h, A, V, text1, R, and various A, B, D labels are used to dynamically visualize and calculate. Suitable for high school students and math enthusiasts studying solid geometry and optimization problems.

GeoGebra Commands & Parameters

  1. Numeric r(5.9)
  2. Numeric h(10.2)
  3. Numeric A(485.0)
  4. V(r) = ((-π) * r^(3)) + (A * r / 2)
  5. (r, V(r))
  6. Numeric A2(1.0)
  7. Numeric A3(2.0)
  8. Numeric A4(3.0)
  9. Numeric A5(4.0)
  10. Numeric A6(5.0)
  11. Numeric A7(6.0)
  12. Numeric A8(7.0)
  13. Numeric A9(8.0)
  14. A
  15. r
  16. A / (((2 * π) * A2)) - A2
  17. A
  18. A
  19. A
  20. A
  21. A
  22. A
  23. A
  24. A
  25. A
  26. A / (((2 * π) * A3)) - A3
  27. A / (((2 * π) * A4)) - A4
  28. A / (((2 * π) * A5)) - A5
  29. A / (((2 * π) * A6)) - A6
  30. A / (((2 * π) * A7)) - A7
  31. A / (((2 * π) * A8)) - A8
  32. A / (((2 * π) * A9)) - A9
  33. Numeric A10(NaN)
  34. A / (((2 * π) * A10)) - A10
  35. A / (((2 * π) * A11)) - A11
  36. ((-π) * A2^(3)) + (D2 * A2 / 2)
  37. ((-π) * A3^(3)) + (D3 * A3 / 2)
  38. ((-π) * A4^(3)) + (D4 * A4 / 2)
  39. ((-π) * A5^(3)) + (D5 * A5 / 2)
  40. ((-π) * A6^(3)) + (D6 * A6 / 2)
  41. ((-π) * A7^(3)) + (D7 * A7 / 2)
  42. ((-π) * A8^(3)) + (D8 * A8 / 2)
  43. ((-π) * A9^(3)) + (D9 * A9 / 2)
  44. ((-π) * A10^(3)) + (D10 * A10 / 2)
  45. ((-π) * A11^(3)) + (D11 * A11 / 2)
  46. h / r