Question

The ratio between the curved surface area
and the total surface area of a cylinder is
1:2. Find the ratio between the height and the
radius of the cylinder

√ Answer needed from stars/mods.​

Answer 1

Answer :

The required ratio 1 : 1

Explanation :

Question says that,

C. S. A. of a cylinder : T. S. A. of the cylinder = 1 : 2

Find the ratio of radius and height.

We know that,

C. S. A.(curved surface area) of a cylinder : $\sf 2\pi rh$

And, T. S. A.(total surface area) : $\sf 2\pi rh + 2\pi r^2 or, 2\pi r(r + h)$

Given :

$\sf \implies 2\pi rh : 2\pi r(r + h) = 1 : 2$

Finding the ratio of height and radius.

Note :

• ‘r’ denotes radius
• ‘h’ is the height.

According to the question :

$\sf\implies 2\pi rh : 2\pi r(r + h) = 1 : 2$

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

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

$\\ \sf \implies 2h = 1(r + h)$

$\\ \sf \implies 2h = r + h$

$\\ \sf\implies 2h - h = r$

$\\ \sf \implies h = r$

$\sf \therefore radius \: is \: equal \: to \: the \: height$

Now, we need to find the ratio of ‘r’ and ‘h'

Let's assume that,

→ r = h = $x$ units.

Their ratio :

$\implies {\sf {r : h }} = x : x$

$\implies {\sf \dfrac{r}{h}} = \dfrac{\not x}{\not x}$

$\implies {\sf \dfrac{r}{h} = \dfrac{1}{1}}$

$\sf \therefore r : h = 1 : 1$

Answer 2

$\begin{gathered}\begin{gathered}\bf \: Given \:  - \begin{cases} &\sf{A \: cylinder \: in \: which} \\ &\sf{CSA : TSA \: = \: 1 :2 } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: find \:  - \begin{cases} &\sf{height_{cylinder} : r_{cylinder}}  \end{cases}\end{gathered}\end{gathered}$

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

$\begin{gathered}\begin{gathered}\bf Let=\begin{cases} &\sf{r \: be \: the \: radius \: of \: cylinder} \\ &\sf{h \: be \: the \: height \: of \: cylinder} \end{cases}\end{gathered}\end{gathered}$

We know,

$\bullet \: {{ \boxed{{\bold\green{Curved \: Surface \: Area_{(Cylinder)}\: = \:2\pi rh}}}}}$

$\bullet \: {{ \boxed{{\bold\green{Total \: Surface \: Area_{(Cylinder)}\: = \:2\pi r(h +r)}}}}}$

Now,

$\begin{gathered}\bf\red{According \: to \: statement}\end{gathered}$

$\rm :\implies\:\dfrac{ CSA_{cylinder}}{TSA_{cylinder}} = \dfrac{1}{2}$

$\rm :\implies\:\dfrac{2\pi rh}{2\pi r(h +r)} = \dfrac{1}{2}$

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

$\rm :\implies\:2h \: = \: h \: + \: r$

$\rm :\implies\:h \: = \: r$

$\rm :\implies\:\dfrac{h}{r} = 1$

$\rm :\implies \: \large \boxed{ \pink{ \bf \: h : r \: = \: 1 : 1}}$

Hence,

$\large\underline{\boxed{ \boxed{ \purple{ \rm \: height_{cylinder} : r_{cylinder} \: = \: 1 : 1}}}}$

More information :-

Perimeter of rectangle = 2(length× breadth)

Diagonal of rectangle = √(length²+breadth²)

Area of square = side²

Perimeter of square = 4× side

Volume of cylinder = πr²h

T.S.A of cylinder = 2πrh + 2πr²

Volume of cone = ⅓ πr²h

C.S.A of cone = πrl

T.S.A of cone = πrl + πr²

Volume of cuboid = l × b × h

C.S.A of cuboid = 2(l + b)h

T.S.A of cuboid = 2(lb + bh + lh)

C.S.A of cube = 4a²

T.S.A of cube = 6a²

Volume of cube = a³

Volume of sphere = 4/3πr³

Surface area of sphere = 4πr²

Volume of hemisphere = ⅔ πr³

C.S.A of hemisphere = 2πr²

T.S.A of hemisphere = 3πr²