Question

A right circular cylinder and a cone have equal bases and equal heights. If their curved
surface areas are in the ratio 8: 5. show that the ratio between radius of their bases to
their height is 3:4​

Answer 1

$\huge{\underline{\purple{\rm{Solution:}}}}$

$\large{\underline{\red{\rm{Given:}}}}$

• $\rm{The\: ratio\:of\: their\:C.S.A=8:5}$

$\large{\underline{\red{\rm{To\: Find:}}}}$

• $\rm{Show\:the\: ratio\:bw\: radius\:and\: height}$

$\large{\underline{\red{\rm{Formula\: used:}}}}$

$\small{\boxed{\green{\rm{CSA\:of\: cyclinder=2\pi\:r\:h}}}}$

$\small{\boxed{\green{\rm{CSA\:of\:cone=\pi\:r\:l}}}}$

$\small{\underline{\red{\rm{As\:per\:the\: above\: information:}}}}$

$\rm{\implies Ratio=\dfrac{C.S.A\:of\: cyclinder}{C.S.A\:of\:cone}=\dfrac{8}{5}}$

$\rm{\implies Ratio=\dfrac{2\pi\:r\:h}{\pi\:r\:l}=\dfrac{8}{5}}$

$\rm\purple{\implies \dfrac{h}{l}=\dfrac{4}{5}}$

• $\rm{Squaring\:on\:both\: side\:\:here,}$

$\rm{\implies (\dfrac{h}{l})^2=(\dfrac{4}{5})^2}$

$\rm{\implies \dfrac{h^2}{l^2}=\dfrac{16}{25}}$

$\rm{\implies l^2=(\dfrac{25}{16})\:h^2}$

• $\rm\green{\boxed{l^2=h^2+r^2}}$

$\rm{\implies h^2+r^2=(\dfrac{25}{16})\:h^2}$

$\rm{\implies r^2=(\dfrac{25}{16})\:h^2-\:h^2}$

$\rm{\implies r^2=\dfrac{25\:h^2-16\:h^2}{16}}$

$\rm{\implies (\dfrac{r}{h})^2=(\dfrac{3}{4})^2}$

Hence,

$\rm\red{\implies \dfrac{r}{h}=\dfrac{3}{4}}$

Answer 2

$\huge\underline{ \mathrm{ \red{QueS{\pink{tiOn}}}}}$

A right circular cylinder and a cone have equal bases and equal heights. If their curved surface areas are in the ratio 8: 5. show that the ratio between radius of their bases to their height is 3:4

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$\large\underline{ \underline{ \blue{ \bold {AnsweR}}}}$

Ratio of Radius and height = $\bf\:\frac{3}{4}=3:4$

$\huge{ \underline{ \purple{ \bold{ \underline{ \mathrm{ExPlanATiOn }}}}}}$

$\large\underline{ \underline{ \red{ \bold {Given}}}}$

• Ratio of CSA is given = 8:5.

• circular cylinder and cone have same base and equal height.

$\large\underline{ \underline{ \red{ \bold {To \:Find}}}}$

We need to show that the ratio of Radius and height is 3:4.

⠀⠀⠀⠀⠀$\huge\underline{ \underline{ \orange{ \bold{sOluTiOn}}}}$

⠀⠀⠀⠀⠀

• Let the Radius be r and height be h .
• Let the slent height be l.

$\: \mapsto \boxed{ \green {\bf{csa \: of \: cone \: = \pi \: rl}}}$

Ratio = $\bf\:\frac{CSA\: of\: cylinder}{CSA\:of\:cone}$

$\bf\:\mapsto\:\frac{h}{l}=\frac{4}{5}$

⠀⠀⠀⠀⠀squaring on both the sides.

$\bf\:\mapsto \frac{h {}^{2} }{l {}^{2} } = \frac{16}{25} \\ \\ \bf\:\mapsto \: l {}^{2} = (\frac{25}{16} ) h {}^{2} \\ \\ \bf\:\mapsto \: h {}^{2} + {r}^{2} =( \frac{25}{16}) h {}^{2} \\ \\ \bf\:\mapsto \: {r}^{2} = \frac{9}{16} {h}^{2} \\ \\ \bf\:\mapsto \: \frac{ {r}^{2} }{ {h}^{2} } = ( \frac{3}{4} ) {}^{2} \\ \\\mapsto{\boxed{\blue{ \bf\: {\frac{r}{h} = \frac{3}{4} }}}}<strong> </strong>$

⠀⠀

Hence, the ratio of Radius and height =3:4.

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