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Tabung dalam Bola

Jar jari tabung = r = x
tinggi tabung =t =  2y
x² + y²  = R²
y² = R² - x²
x² =R² - y²

a) V tabung = π r² t = π (R²-y²)(2y)
V tabung = 2 πy (R²-y²)

b) t maksimum = ...
t = 2y
.
V tab = 2 π y (R² - y²)
V(y) = 2 π R²  y - 2 π y³
V' (y)= 0 --> 2 π R² - 6 πy² = 0
2πR² = 6 π y²
R² = 3 y² --> y² = R²/3
y = √(R²/3) = R/√3
t = 2y -->
t = 2R/√3
.
c. V tabung : V bola
V tabung = π r² t
Vtab = π (R²-y²)(2R/√3)
V tab = π (R² - R²/3) (2R)(1/√3)
V tab = π (2/3  R²)(2R)(1/3)(√3)
V tab = π R³ (4/9)(√3)
Vbola  = π R³ (4/3)
.
V tab : Vbola = (4/9)√3  : (4/3) =  (4√3) : (12) = √3 : 3 = 1 : √3