Answer

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**Hint:**We know that the formula of total surface area and curved surface area of a cylinder are $TSA = 2\pi r\left( {h + r} \right)sq.units$and $CSA = 2\pi rhsq.units$ respectively. And since they are in a ratio of 1:2 we can write them as $\dfrac{{2\pi rh}}{{2\pi r\left( {h + r} \right)}} = \dfrac{1}{2}$and further simplifying we get that the radius is equal to the height and using this the TSA formula and equation with the given TSA we get the value of r and we can find the volume using the formula $ \Rightarrow Volume = \pi {r^2}h{\text{ }}cu.units$.

**Complete step by step solution:**

We are given the total surface area of a solid cylinder to be 616 $c{m^2}$

We know that the formula for the total surface area of the cylinder is

$ \Rightarrow TSA = 2\pi r\left( {h + r} \right)sq.units$

And the formula for the curved surface area of the cylinder is

$ \Rightarrow CSA = 2\pi rhsq.units$

We are given that the ratio between the total surface area and curved surface area is 1 : 2

$

\Rightarrow \dfrac{{2\pi rh}}{{2\pi r\left( {h + r} \right)}} = \dfrac{1}{2} \\

\Rightarrow \dfrac{h}{{\left( {h + r} \right)}} = \dfrac{1}{2} \\

\Rightarrow 2h = h + r \\

\Rightarrow 2h - h = r \\

\Rightarrow h = r \\

$

We can use this the total surface area formula to find the value of r

Given total surface area is 616 $c{m^2}$

$ \Rightarrow 616 = 2\pi r\left( {h + r} \right)$

Since h = r

\[

\Rightarrow 616 = 2\pi r\left( {r + r} \right) \\

\Rightarrow 616 = 2\pi r\left( {2r} \right) \\

\Rightarrow 616 = 4\pi {r^2} \\

\Rightarrow \dfrac{{616}}{4}\times \dfrac{7}{{22}} = {r^2} \\

\Rightarrow \dfrac{{56}}{4}\times \dfrac{7}{2} = {r^2} \\

\Rightarrow 7\times 7 = {r^2} \\

\Rightarrow r = 7cm \\

\]

Hence now we get the radius of the cylinder to be 7 cm

And since the height and radius are equal the height is also 7 cm

We know that the volume of the cylinder is given by

$ \Rightarrow Volume = \pi {r^2}h{\text{ }}cu.units$

Substituting the values we get

$

\Rightarrow Volume = \dfrac{{22}}{7}\times 7\times 7\times 7 \\

\Rightarrow Volume = 22\times 49 = 1078c{m^3} \\

$

**Hence the volume of the cylinder is $1078c{m^3}$.**

**Note :**

Many students tend to use the given TSA while equating with the ratios . It will only make the process tedious.

A cylinder is one of the most basic curved geometric shapes, with the surface formed by the points at a fixed distance from a given line segment, known as the axis of the cylinder. The shape can be thought of as a circular prism. Both the surface and the solid shape created inside can be called a cylinder.

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