A cylindrical vessel open at the top has a base diameter \[{\mathbf{56}}\]cm if the total cost of painting the outer curved surface of the vessel is \[{\mathbf{352}}\] rupee at the rate rupee \[{\mathbf{0}}.{\mathbf{2}}\] per \[{\mathbf{100}}\]cm square then the height of the vessel is

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Hint: A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance. These bases are normally circular in shape and the center of the two bases are joined by a line segment, which is called the axis. The perpendicular distance between the bases is the height, “h” and the distance from the axis to the outer surface is the radius “r” of the cylinder.

Curved Surface Area: The area of the curved surface of the cylinder which is contained between the two parallel circular bases. It is also stated as a lateral surface area. The formula for it is given by:

\[{\text{surface area of cylinder = 2}}\pi {\text{rh}}\]

Complete answer:

First of all we will find the area of the curved surface:
\[{\text{Rs }}0.{\text{2}}0{\text{ = 1}}00cm{.^2}\]
\[{\text{Rs 1 }} = {\text{ }}\dfrac{{100}}{{0.2}} = {\text{ 5}}00c{m^2}\]
\[{\text{Rs 352 }} = {\text{ 5}}00{\text{ x 352 }} = {\text{ 176}}000c{m^2}\]

Then we will find the radius:
\[{\text{Radius }} = {\text{ }}\dfrac{{{\text{Diameter}}}}{2}\]
\[{\text{Radius }} = {\text{ }}\dfrac{{56}}{2} = 28cm\]

then we will find the height:
\[{\text{Curved surface area }} = {\text{ 2}}\pi h\]
Given that area = \[\,{\text{176}}00c{m^2}\] and the radius = \[{\text{2}}cm.\]
\[{\text{2}}\pi \left( {{\text{28}}} \right)h{\text{ }} = {\text{ 176}}000\]
\[{\text{56}}\pi h{\text{ }} = {\text{ 176}}000\]
\[h = {\text{ }}\dfrac{{{\text{176}}000}}{{56\pi }}\]
\[h = {\text{1}}000{\text{ cm or 1}}0{\text{m}}\]

Hence, the height is \[{\text{1}}0\]m

Note: LPG gas-cylinder is one of the real-life examples. It is a three-dimensional shape having surface area and volume. The total area of the cylinder is equal to the sum of its curved surface area and area of the two circular bases.
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