The radii of the internal and external surfaces of a metallic spherical shell are 3 cm and 5 cm respectively. It is melted and recast into a solid right circular cone of height 32 cm. Find the diameter of the base of the cone.

Given,

Height of the solid right circular cone (h) = 32 cm

Internal radius of metallic spherical shell (r) = 3 cm

External radius of metallic spherical shell (R) = 5 cm

Let radius of cone be r₁ cm.

As, metallic spherical shell is recasted into right circular cone.

∴ Volume of spherical shell = Volume of cone.

$\Rightarrow \dfrac{4}{3}π(R^3 - r^3) = \dfrac{1}{3}πr$1^2h \\[1em] \Rightarrow 4(R^3 - r^3) = r1^2h \\[1em] \Rightarrow r1^2 = \dfrac{4(R^3 - r^3)}{h} \\[1em] \Rightarrow r1^2 = \dfrac{4 \times (5^3 - 3^3)}{32} \\[1em] \Rightarrow r1^2 = \dfrac{4 \times (125 - 27)}{32} \\[1em] \Rightarrow r1^2 = \dfrac{4 \times 98}{32} \\[1em] \Rightarrow r1^2 = \dfrac{49}{4} \\[1em] \Rightarrow r1 = \sqrt{\dfrac{49}{4}} \\[1em] \Rightarrow r_1 = \dfrac{7}{2}.

Diameter = 2r = $2 \times \dfrac{7}{2}$ = 7 cm.

Hence, diameter = 7 cm.