Question

Two cones have their heights in the ratio 1 : 3 and the radius of their bases in the ratio 3 : 1, show that their volumes are in the ratio 3 : 1​

Answer 1

$\large\bold{\underline{\underline{Question:-}}}$

Two cones have their heights in the ratio 1 : 3 and the radius of their bases in the ratio 3 : 1 show that their volumes are in the ratio 3 : 1.

$\large\bold{\underline{\underline{Solution:-}}}$

⇒ Ratio of heights of two cones = 1 : 3

⇒ Ratio of radius of their bases = 3 : 1

⇒ We know that V = 1/3πr²h

⇒ Ratio of their volume = V1 : V2

$\underline{ \underline{ \sf{Let's \: find \: the \: ratio \: of \: their \: volumes:}}}$

${ \implies{ \sf {V _{1} : V_{2}}}}$

${ \implies{ \sf{ \frac{1}{3} \pi \: { r_{1}}^{2} h_{1} : \frac{1}{3} \pi \: { r_{2} }^{2} h_{2}}}}$

${ \implies{ \sf{ {r_{1}}^{2} h_{1} : { r_{2}}^{2} h_{2}}}}$

${ \implies{ \sf{ \frac{ { r_{1} }^{2} }{ { r_{2} }^{2} } : \frac{ h_{2} }{ h_{1} } }}}$

${ \implies{ \sf{ \frac{ {3}^{2} }{ {1}^{2} } : \frac{3}{1} }}}$

${ \implies{ \sf{ \frac{9}{1} : \frac{3}{1} }}}$

${ \implies{ \sf{3 : 1}}}$

${ \therefore{ \sf{ \green{Ratio \: of \: their \: volumes = 3 : 1}}}}$