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Parallelepipedo numero 74

Α(lat) = 5·√3/8·pi·L^2

pi·(L + r)·a = 5·√3/8·pi·L^2

α = pi/2 - x

h = a·SIN(α)

r/h = TAN(α)-----> r = h·TAN(α)

a = l·SIN(x)

r = (a·SIN(α))·TAN(α)---> r = a·SIN(α)^2/COS(α)

r = (L·SIN(x))·SIN(α)^2/COS(α)

r = (L·SIN(x))·SIN(pi/2 - x)^2/COS(pi/2 - x)

r = L·COS(x)^2

(L + L·COS(x)^2)·(L·SIN(x)) = 5·√3/8·L^2

L^2·SIN(x)·COS(x)^2 + L^2·SIN(x) = 5·√3/8·L^2

SIN(x)·COS(x)^2 + SIN(x) = 5·√3/8

pongo:

COS(x) = Χ

SIN(x) = Υ

Risolvo:

{Υ·Χ^2 + Υ = 5·√3/8

{Χ^2 + Υ^2 = 1

ottengo:

[Υ = √3/2 ∧ Χ = 1/2, Υ = (√23- √3)/4 ∧ Χ = √(2·√69 - 10)/4]

Quindi soluzioni:

x = pi/3 v x = ASIN((√23 - √3)/4)