Sphere and cone

Within the sphere of radius G = 33 cm inscribe the cone with the 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 c m V = \frac{1}{3} π r^{2} h h = G + x = 33 + x G^{2} = x^{2} + r^{2} r^{2} = G^{2} - x^{2} V = \frac{1}{3} π (G^{2} - x^{2}) (G + x) V = \frac{1}{3} π (G^{3} + G^{2} x - G x^{2} - x^{3}) V^{'} = \frac{1}{3} π (G^{2} - 2 G x - 3 x^{2}) V^{'} = 0 G^{2} - 2 G x - 3 x^{2} = 0 - 3 x^{2} - 66 x + 1089 = 0 3 x^{2} + 66 x - 1089 = 0 a = 3; b = 66; c = - 1089 D = b^{2} - 4 a c = 6 6^{2} - 4 \cdot 3 \cdot (- 1089) = 17424 D > 0 x_{1, 2} = \frac{- b \pm \sqrt{D}}{2 a} = \frac{- 66 \pm \sqrt{17424}}{6} x_{1, 2} = \frac{- 66 \pm 132}{6} x_{1, 2} = - 11 \pm 22 x_{1} = 11 x_{2} = - 33 Factored form of the equation: 3 (x - 11) (x + 33) = 0 h = G + x_{1} = 33 + 11 = 44 c m r = \sqrt{G^{2} - x_{1}^{2}} = 31.11 cm

h = 33 + 11 = 44 cm

V = \frac{1}{3} π r^{2} h = 44602.24 {cm}^{3}

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Showing 1 comment:

Dr Math

that's very mind blowing

2 years ago Like

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