# Cone

Into rotating cone with dimensions r = 8 cm and h = 8 cm incribe cylinder with maximum volume so that the cylinder axis is perpendicular to the axis of the cone. Determine the dimensions of the cylinder.

Correct result:

r =  2.67 cm
h =  5.33 cm

#### Solution:

$V = \dfrac13 \pi r^2 h \ \\ x/R = 2r/H \ \\ \ \\ r = \dfrac{H }{ 2R } x = \dfrac{ 8 }{ 2 \cdot 8 } x \ \\ h = 2(R-x) = 2(8-x) \ \\ \ \\ V = \dfrac13 \pi (\dfrac{ 8 }{ 2 \cdot 8 } x )^2 \cdot 2(8-x) \ \\ \ \\ V = \dfrac{ \pi 8^2}{ 6 \cdot 8^2 } ( 8 x^2- x^3) \ \\ \ \\ V = \dfrac{ \pi 8^2}{ 6 \cdot 8^2 } ( 2\cdot 8 x - 3x^2) \ \\ V = 0 \ \\ 2\cdot 8 x - 3x^2 = 0 \ \\ x_1 = 0 \ \\ 2\cdot 8 - 3x = 0 \ \\ \ \\ 16 = 3x \ \\ \ \\ x = 5.333 \ cm \ \\ r = \dfrac{ 8 }{ 2 \cdot 8 } \cdot 5.333 = 2.67 \ \text{cm} \ \\$
$h = 2(8-5.333) = 5.33 \ \text{cm}$

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