Question

the radii of two cylinder are in ratio of 1 is to 3 if the volume of two cylinder with same find the ratio of their respective Heights​

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\pink{\underline{\mathfrak{\bf{Given}}}}}$

• The radii of two cylinder are in ratio = 1:3
• The volume of two cylinder with same

$\Large{\pink{\underline{\mathfrak{\bf{Find}}}}}$

• The ratio of their respective Heights

$\Large{\underline{\underline{\mathfrak{\bf{Explanation}}}}}$

Let,

• Radius of first cylinder = r
• Radius of Second cylinder = r'
• Height of First cylinder = h
• Height of second cylinder = h'

A/C to queation:-

(The radii of two cylinder are in ratio = 1:3)

$\small\boxed{\sf{\orange{\:r:r'\:=\:1:3\:=\:\dfrac{r}{r'}\:=\:\dfrac{1}{3}}}} \\ \\ \implies\sf{\red{\:3r\:=\:r'.......(1)}}$

we know,

$\small\boxed{\sf{\orange{\:Volume_{cyli der}\:=\:\pi.(Radius)^2.(Height)}}}$

$\bigstar\sf{\green{\:volume_{first\:cylender}\:=\:\pi.r^2.h}} \\ \\ \bigstar\sf{\green{\:volume_{second\:cyli der}\:=\:\pi.r'^2.h'}}$

Again,A/C to question

(The volume of two cylinder with same )

$\small\boxed{\sf{\orange{\:Volume_{First\:cylinder}\:=\:Volume_{second\:cylinder}}}} \\ \\ \implies\sf{\:\pi.r^2.h\:=\:\pi.r'^2.h'} \\ \\ \implies\sf{\:\dfrac{h}{h'}\:=\:\dfrac{\cancel{\pi}.r'^2}{\cancel{\pi}.r^2.}} \\ \\ \implies\sf{\:\dfrac{h}{h'}\:=\:\dfrac{r'^2}{r^2}}$

Keep value by equ(1),

• r' = 3r

$\implies\sf{\:\dfrac{h}{h'}\:=\:\dfrac{(2r)^2}{r^2}} \\ \\ \mapsto\sf{\:\dfrac{h}{h'}\:=\:\dfrac{9\cancel{r^2}}{\cancel{r^2}}} \\ \\ \implies\sf{\:\dfrac{h}{h'}\:=\:9}$

So, we can write this , in ratio

$\implies\small\boxed{\sf{\orange{\:h:h'\:=\:4:1}}}$

$\Large{\pink{\underline{\mathfrak{\bf{Thus}}}}}$

• Ratio of height of two cylinder (h:h')= ( 9:1)