# Sphere and cone

Within the sphere of radius G = 33 cm inscribe cone with largest volume. What is that volume and what are the dimensions of the cone?

Correct result:

r =  31.11 cm
h =  44 cm
V =  44602.24 cm3

#### Solution:

$G = 33 \ cm \ \\ V = \dfrac13 \pi r^2 h \ \\ h = G + x = 33 +x \ \\ G^2 = x^2+r^2 \ \\ r^2 = G^2 - x^2 \ \\ V = \dfrac13 \pi (G^2 - x^2)(G+x) \ \\ V = \dfrac13 \pi (G^3 + G^2x - Gx^2 - x^3) \ \\ \ \\ V' = \dfrac13 \pi (G^2 - 2Gx - 3x^2) \ \\ V'=0 \ \\ G^2 - 2Gx - 3x^2=0 \ \\ -3x^2 -66x +1089 =0 \ \\ 3x^2 +66x -1089 =0 \ \\ \ \\ a=3; b=66; c=-1089 \ \\ D = b^2 - 4ac = 66^2 - 4\cdot 3 \cdot (-1089) = 17424 \ \\ D>0 \ \\ \ \\ x_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ -66 \pm \sqrt{ 17424 } }{ 6 } \ \\ x_{1,2} = \dfrac{ -66 \pm 132 }{ 6 } \ \\ x_{1,2} = -11 \pm 22 \ \\ x_{1} = 11 \ \\ x_{2} = -33 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 3 (x -11) (x +33) = 0 \ \\ \ \\ \ \\ h = G + x_1 = 33 + 11 = 44 \ cm \ \\ r = \sqrt{ G^2 - x_1^2 } = 31.11 \ \text{cm} \ \\$

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Showing 1 comment:
Dr Math
that's very mind blowing

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