Question

the ratio between the curved surface area and the total surface area of right circular cylinder are in ratio 1:2 . find the ratio between the height and radius of the cylinder .​

Answer 1

† Question †

The ratio between the curved surface area and the total surface area of right circular cylinder are in ratio 1:2 . find the ratio between the height and radius of the cylinder .

$\rule{300}2$

$\implies\huge\frac {c.s.a-of-cylinder}{T.S.A-of-cylinder}=\frac{1}{2}$

TO FIND,

★ the ratio between the height and radius of the cylinder =?

$\implies\huge\frac{h}{r}=?$

◇ Solution ◇

Let,

→ The radius of cylinder = "r"

→ height of cylinder = "h"

→ C.S.A of cylinder = 2πrh

→ T.S.A of cylinder = 2πrh+π$r^2$

$\rule{150}2$

★ So ★

$\implies\frac{2πrh}{2πrh+πr^2}=\frac{1}{2} \\ \implies \frac{2πrh}{2πr (h+r)}=\frac{1}{2}\\ \implies \frac{h}{h+r}=\frac{1}{2}\\ \implies 2h=h+r\\ \implies h=r\\ \implies \frac{h}{r}=\frac{1}{1}\\ \implies {\fbox{\fbox{h:r=1:1}}}$

$\rule{300}2$

Hence,

→ the ratio between the height and radius of the cylinder$\implies\huge\red{\fbox{\fbox{h:r=1:1}}}$

Answer 2

Answer- The above question is from the chapter 'Surface Areas and Volumes'.

Concept used: 1) CSA of a cylinder = 2πrh

2) TSA of a cylinder = 2πr(r+h)

Given question: The ratio between the curved surface area and the total surface area of right circular cylinder are in ratio 1:2 . Find the ratio between the height and radius of the cylinder.

Solution: Let r be the radius and h be the height of a right circular cylinder.

It is given that CSA:TSA = 1:2

2πrh : 2πr(r+h) = 1 : 2

$\dfrac{2 \pi r h}{2 \pi r(r+h)}$ = $\dfrac{1}{2}$

$\dfrac{h}{r \: + \: h} = \dfrac{1}{2}$

On cross-multiplying, we get,

2h = r + h

Transposing h to L.H.S, we get,

2h - h = r

h = r

⇒ Radius and height are equal.

Ratio of radius and height = $\frac{r}{h}$

                                             = $\frac{r}{r}$ (∵ r = h)

                                             = $\frac{1}{1}$

∴ Required ratio = 1 : 1.