Question

A solid metallic sphere of diameter 21cm is melted and recast into a number of cones,each of diameter 7cm and height 3cm.The number of cones so formed is

Answer 1

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{126 \ cones \ are \ formed.}$

$\sf\orange{Given:}$

$\sf{For \ sphere,}$

$\sf{\implies{Diameter (d)=21 \ cm}}$

$\sf{For \ cone,}$

$\sf{\implies{Diameter (D)=7 \ cm}}$

$\sf{\implies{Height (h)=3 \ cm}}$

$\sf\pink{To \ find:}$

$\sf{The \ number \ of \ cones \ formed.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{For \ sphere,}$

$\sf{Radius (r)=\frac{Diameter}{2}=\frac{21}{2}}$

$\sf{\therefore{Radius (r)=\frac{21}{2} \ cm}}$

$\sf{For \ cone,}$

$\sf{Radius (R)=\frac{Diameter}{2}=\frac{7}{2}}$

$\sf{\therefore{Radius (R)=\frac{7}{2} \ cm}}$

$\boxed{\sf{Volume \ of \ sphere=\frac{4}{3}\times\pi\times \ r^{3}}}$

$\boxed{\sf{Volume \ of \ cone=\frac{1}{3}\times\pi \ r^{2}\times \ h}}$

$\sf{\therefore{Number \ of \ cones \ formed}}$

$\sf{=\frac{Volume \ of \ sphere}{Volume \ of \ cone}}$

$\sf{=\frac{\frac{4}{3}\times\pi\times \ r^{3}}{\frac{1}{3}\times\pi\times \ R^{2}\times \ h}}$

$\sf{=\frac{4\times \ r^{3}}{R^{2}\times \ h}}$

$\sf{=\frac{4\times(\frac{21}{2})^{3}}{(\frac{7}{2})^{2}\times3}}$

$\sf{=\frac{21\times21\times21\times4}{2\times7\times7\times3}}$

$\sf{=21\times3\times2}$

$\sf{=21\times6}$

$\sf{\therefore{Number \ of \ cones \ formed=126}}$

$\sf\purple{\tt{\therefore{126 \ cones \ are \ formed.}}}$