Question

Question 9
A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form of a cone of base diameter 8cm. Find the height of the cone.
(A) 12 cm
(B) 14 cm
(C) 15 cm
(D) 18 cm

Answer 1

Option (B) is correct.
Given, internal diameter of spherical shell = 4 cm
External diameter of spherical = 8 cm
$\therefore$ internal radius of spherical shell, $r_1=\frac{4}{2}cm=2cm[Diameter=2\times radius]$
External radius of shell, $r_2=\frac{8}{2}=4cm$ $[Diameter=2\times radius]$

Given is the spherical shell
Now , volume of the spherical shell = $\frac{4}{3}\pi [r_2^3-r_1^3]$
$[\because$ Volume of the spherical shell $=\frac{4}{3}\pi \left(externalradiu{s}^3\right)-(internalradius{)}^3]=\frac{4}{3}\pi (4^3-2^3)=\frac{4}{3}\pi (64-8)=\frac{224}{3}\pi c{m}^3$
Let height of the cone = h cm
Diameter of the base of cone = 8 cm $[diameter=2\times radius]$
According to the question,
Volume of cone = Volume of spherical shell $[Volumeofcone=\frac{1}{3}\times \pi \times (radius{)}^2\times \left(height\right)]$
$\Rightarrow \frac{1}{3}\pi (4{)}^{2}h=\frac{224}{3}\pi \Rightarrow h=\frac{224}{16}=14cm$
Hence, the height of the cone is 14 cm.