Question

A metal pipe is \[77cm\] long. The inner diameter of a cross-section is \[4{\text{ }}cm,\] the outer diameter being then found its total surface area.

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Answer 1

Hint: To calculate the total surface area we will add inner and outer curved surface area and also the area of both the ends.


Complete step by step solution:
We are given that:

A pipe will have two layers. There is an inner cylinder of pipe and outer cylinder of pipe.

Inner radius of pipe \[ = \dfrac{{InnerDiameter}}{2} = \dfrac{4}{2} = 2cm = r\]

Outer radius of pipe \[ = \dfrac{{OuterDiameter}}{2} = \dfrac{{4.4}}{2} = 2.2cm = R\]

Height of the pipe \[ = {\text{ }}77cm{\text{ }}\left( {Given} \right){\text{ }} = {\text{ }}h\]

Inner curved surface area of pipe \[ = 2\pi rh = 2 \times \dfrac{{22}}{7} \times 2 \times 77 = 968c{m^2}\]

Outer curved surface area of the pipe \[ = \;2\pi Rh = 2 \times \dfrac{{22}}{7} \times 2.2 \times 77 = 1064.8\]

Area of one end of cylinder \[ = \pi \left( {{R^2} - {r^2}} \right)\]

          \[\;\;\;\;\;\; = 2 \times \dfrac{{22}}{7} \times \left( {2.2 \times 2.2 - 2 \times 2} \right)\]

               \[\;\;\;\;\;\; = {\text{ }}5.28c{m^2}\]

\[\left( {This{\text{ }}is{\text{ }}the{\text{ }}area{\text{ }}between{\text{ }}two{\text{ }}concentric circles} \right)\]

Total Surface area \[ = \] Inner surface area \[ + \] Outer surface area \[ + \]Area of both ends of cylinder

\[ = {\text{ }}968 + 1064.8 + 2 \times 5.28\]
\[ = {\text{ }}2043.36c{m^2}\]

$\therefore $Total surface area of pipe is \[2043.36c{m^2}\]


Note: To find area of ends of the cylinder \[\left( {area{\text{ }}between{\text{ }}concentric{\text{ }}circles} \right)\] we find difference between the areas of outer and inner circles.