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Formula expansion ratio

a guest Mar 28th, 2020 98 Never
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  1. L = cte //Longitud
  2. n = cte //numero de celdas
  3. ratio = cte //expansion ratio
  4. sum c0...cn-1 = L
  5.  
  6. Como indexamos desde 0, va hasta n-1
  7. c0 = l
  8. c1 = cratio * c0
  9. c2 = cratio * c1
  10. .
  11. .
  12. .
  13. cn-1 = cratio * cn-2 = ratio*c0
  14.  
  15. cn-1 = cratio^2 * cn-3 = ratio*c0
  16. (reemplazando para atras)
  17. cn-1 = cratio^(n-1) * cn-n = ratio*c0
  18. cn-1 = cratio^(n-1) * c0 = ratio * c0
  19. cratio^(n-1) = ratio
  20. cratio = ratio^(1/(n-1))
  21.  
  22. Ademas sum c0...cn-1 = L
  23. sum(cratio^i * c0) = L
  24. c0 sum(cratio^i) = L
  25. https://en.wikipedia.org/wiki/Geometric_series#Sum
  26. c0 (1 - cratio^n)/(1 - cratio) = L
  27. c0 = L * (1 - cratio) / (1 - cratio^n)
  28.  
  29.  
  30. Para nuestro caso
  31.  
  32. L = 0.08
  33. n = 800
  34. ratio = 10
  35. cratio = 10^(1/799) ~ 1.00288599
  36. c0 ~ 0.08 * (1- 1.00288599) / (1 - 1.00288599^800) = 0.0000255712
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