A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form a cone of base diameter 8 cm. The height of the cone is ______.

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A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form a cone of base diameter 8cm. The height of the cone is 14 cm. Explanation: Volume of spherical shell = Volume of cone recast by melting For spherical shell, Internal diameter, d1 = 4 cm Internal radius, r1 = 2 cm ...[As radius = $\frac{1}{2}$ diameter] External diameter, d2 = 8 cm External radius, r2 = 4 cm Now, As volume of spherical shell= $\frac{4}{3}$ $\pi$("r"_2^3 - "r"_1^3) Where r1 and r2 are internal and external radii respectively. Volume of given shell = $\frac{4}{3}$ $\pi$(4^3 - 2^3) = $\frac{4}{3}$ $\pi$(56) = ($\frac{224}{3}$)$\pi$ We know that, Volume of cone = (224$\pi$)/3 "cm"^3 For cone, Base diameter = 8 cm Base radius, r = 4 cm Let Height of cone = ‘h’. We know, Volume of cone = ($\frac{1}{3}$)$\pi$ "r"^2"h", Where r = Base radius and h = Height of cone Volume of given cone = ($\frac{1}{3}$)$\pi$4^2"h" $\Rightarrow$ (224$\pi$)/3 = (16$\pi$"h")/3 $\Rightarrow$ 16h = 224 h = 14 cm So, Height of cone is 14 cm. [Solution diagram: 268785_solution_1.png]

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