The radii of the bases of two right circular solid cones of the same height are r1 & r2. The cones are melted & recasted into a solid sphere if radius R . show that the height of each cone is given by h=4R^3÷r1^2+r2^2 ?

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The radii of the bases of two right circular solid cones of the same height are r1 & r2. The cones are melted & recasted into a solid sphere if radius R . show that the height of each cone is given by h=4R^3÷r1^2+r2^2 ?

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Answer 1 · 2018-02-27T18:52:36

See below. Quite simple really.

Explanation:

Volume of cone 1; $π \cdot r_{1}^{2} \cdot \frac{h}{3}$ $pi*r_1^2*h/3$
Volume of cone 2: $π \cdot r_{2}^{2} \cdot \frac{h}{3}$ $pi*r_2^2*h/3$
Volume of the sphere: $\frac{4}{3} \cdot π \cdot r^{3}$ $4/3*pi*r^3$

So you have:

$Vol of sphere = Vol of cone 1 + Vol of cone 2$ $"Vol of sphere" = "Vol of cone 1" + "Vol of cone 2"$

$\frac{4}{3} \cdot π \cdot R^{3} = (π \cdot r_{1}^{2} \cdot \frac{h}{3}) + (π \cdot r_{2}^{2} \cdot \frac{h}{3})$ $4/3*pi*R^3 = ( pi*r_1^2*h/3 ) + (pi*r_2^2*h/3)$

Simplify:

$4 \cdot π \cdot R^{3} = (π \cdot r_{1}^{2} \cdot h) + (π \cdot r_{2}^{2} \cdot h)$ $4*pi*R^3 = (pi*r_1^2*h ) + (pi*r_2^2*h)$

$4 \cdot R^{3} = (r_{1}^{2} \cdot h) + (r_{2}^{2} \cdot h)$ $4*R^3 = ( r_1^2*h ) + (r_2^2*h)$

$h = \frac{4 R^{3}}{r_{1}^{2} + r_{2}^{2}}$ $h = (4R^3)/ ( r_1^2 + r_2^2)$

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The radii of the bases of two right circular solid cones of the same height are r1 & r2. The cones are melted & recasted into a solid sphere if radius R . show that the height of each cone is given by h=4R^3÷r1^2+r2^2 ?

1 Answer

Explanation:

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