Question

Prove that ratio of volume of cube to volume of sphere is 6/pi

Answer 1

$\sf\blue{Correct \ question:}$

$\sf{A \ cube \ and \ a \ sphere \ has \ same \ height.}$

$\sf{Prove \ that \ ratio \ of \ volume \ of \ cube}$

$\sf{to \ volume \ of \ sphere \ is \ 6:\pi}$

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$\sf\red{\underline{\underline{To \ prove:}}}$

$\sf{Ratio \ of \ volume \ of \ cube \ to \ volume}$

$\sf{of \ sphere \ is \ 6:\pi}$

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

$\sf{For \ cube,}$

$\sf{All \ dimensions \ are \ equal}$

$\sf{\therefore{Side(l)=Height (h)}}$

$\sf{For \ sphere,}$

$\sf{Diameter (d)=Height (h)}$

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

$\sf{\therefore{Radius(r)=\frac{Height (h)}{2}}}$

$\boxed{\sf{Volume \ of \ cube=l^{3}}}$

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

$\sf{Ratio \ of \ volume \ of \ cube \ and \ sphere}$

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

$\sf{=\frac{h^{3}}{\frac{4}{3}\times\pi\times(\frac{h}{2})^{3}}}$

$\sf{=\frac{h\times \ h\times \ h}{\frac{4}{3}\times\pi\times\frac{h}{2}\times\frac{h}{2}\times\frac{h}{2}}}$

$\sf{=\frac{h\times \ h\times \ h\times2\times2\times2\times3}{4\times\pi\times \ h\times \ h\times \ h}}$

$\sf{=\frac{2\times3}{\pi}}$

$\sf{=\frac{6}{\pi}}$

$\sf{\therefore{Ratio \ of \ volumes \ of \ cube \ and}}$

$\sf{sphere=6:\pi}$

$\sf\purple{\tt{Hence, \ proved}}$

$\sf\purple{\tt{Ratio \ of \ volume \ of \ cube \ to}}$

$\sf\purple{\tt{volume \ of \ sphere \ is \ 6:\pi.}}$