Question

The radii of the bases of three right circular metallic cones of same height 'h' are 8 cm, 16 cm and 24 cm. The
cones are melted together and recast into a solid sphere of radius 28 cm. Find the value of 'h'.
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Answer 1

$\textbf{Given:}$

$\textsf{Radii of three metallic cones having height 'h' are 8cm, 16cm and 24 cm}$

$\textsf{Radius of the solid sphere is 28 cm}$

$\textbf{To find:}$

$\textsf{The value of 'h'}$

$\textbf{Solution:}$

$\textbf{Formula used:}$

$\boxed{\mathsf{Volume\;of\;cone=\dfrac{1}{3}\pi\,r^2\,h\;cubic\,units}}$

$\boxed{\mathsf{Volume\;of\;sphere=\dfrac{4}{3}\pi\,r^3\,cubic\,units}}$

$\textsf{As per given data,}$

$\textsf{Volume of three cones=Volume of the sphere}$

$\implies\mathsf{\dfrac{1}{3}\pi(8)^2\,h+\dfrac{1}{3}\pi(16)^2\,h+\dfrac{1}{3}\pi(24)^2\,h=\dfrac{4}{3}\pi\,(28)^3}$

$\implies\mathsf{\dfrac{1}{3}\pi\,h[8^2+16^2+24^2]=\dfrac{4}{3}\pi\,(28)^3}$

$\implies\mathsf{h[64+256+576]=4{\times}28{\times}28{\times}28}$

$\implies\mathsf{h{\times}896=4{\times}28{\times}28{\times}28}$

$\implies\mathsf{h=\dfrac{4{\times}28{\times}28{\times}28}{896}}$

$\implies\mathsf{h=\dfrac{28{\times}28{\times}28}{224}}$

$\implies\mathsf{h=\dfrac{28{\times}28}{8}}$

$\implies\mathsf{h=\dfrac{7{\times}28}{2}}$

$\implies\mathsf{h=7{\times}14}$

$\implies\boxed{\mathsf{h=98\;cm}}$

$\textbf{Answer:}$

$\textsf{Height of the sphere is 98 cm}$