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# The total surface area of a solid cylinder is 616 $c{m^2}$. If the ratio between its curved surface area and total surface area is 1:2. Find the volume of the cylinder?

Last updated date: 10th Aug 2024
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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.