Question

A metal pipe is 77cm long. The inner diameter of a cross section is 4cm, the outer diameter being 4.4cm. Find its
$
  ({\text{i}}){\text{ inner curved surface area}} \\
  ({\text{ii}}){\text{ outer curved surface area}} \\
  ({\text{iii}}){\text{ total surface area}} \\
 $

Answer 1

Hint:- In this question first we need to calculate the inner radius and the outer radius because in question diameters of both cross sections are given. Then we have to put in respective formulas to get the answer. And for total surface area we need to add inner, outer curved surface area and two times the area of base.

Complete step-by-step answer:

Given: A metal pipe is 77cm long. Let its height be (h)=77cm.
The inner diameter of the cross section is 4cm. Let it be ${r_1}$=2cm {radius=$\dfrac{{{\text{diameter}}}}{2}$}
The outer diameter of the cross section is 4.4cm. Let it be ${r_2}$=2.2cm .
We know curved surface area of cylinder=$2\pi rh$
Then, the curved surface area of inner surface of pipe=$2\pi {r_1}h$
=$\left( {2 \times \dfrac{{22}}{7} \times 2 \times 77} \right)$ ${\text{ }} \Rightarrow {\text{take }}\pi {\text{ = }}\dfrac{{22}}{7}$
 the curved surface area of inner surface of pipe = $968c{m^2}$


Now,
We know curved surface area of cylinder=$2\pi rh$
Then, the curved surface area of inner surface of pipe=$2\pi {r_2}h$
=\[\;\left( {2 \times \dfrac{{22}}{7} \times 2.2 \times 77} \right)\] ${\text{ }} \Rightarrow {\text{take }}\pi {\text{ = }}\dfrac{{22}}{7}$
the curved surface area of inner surface of pipe = $1064.8c{m^2}$

Now,
We know , Total surface area = Curved surface area of inner cylinder + curved surface area of outer cylinder + 2$ \times $Area of base
Area of base = Area of circle with radius 2.2 cm - Area of circle with radius 2cm
 =$\pi {r_2}^2 - \pi r_1^2$
 =$\dfrac{{22}}{7} \times \{ {(2.2)^2} - {(2)^2}\} {\text{ \{ take }}\pi {\text{ = }}\dfrac{{22}}{7}\} $
=$2.74c{m^2}$
Then Total surface area = $968{\text{ }} + {\text{ }}1064.8{\text{ }} + {\text{ }}2 \times 2.74$
=$2038.08c{m^2}$
Hence,
$
  ({\text{i}}){\text{ inner curved surface area = 968c}}{{\text{m}}^2} \\
  ({\text{ii}}){\text{ outer curved surface area = 1064}}{\text{.8c}}{{\text{m}}^2} \\
  ({\text{iii}}){\text{ total surface area = 2038}}{\text{.08c}}{{\text{m}}^2} \\
$

Note:- Whenever you get this type of question the key concept to solve this is to learn all the formulas related to the shape like in this question we require the formula of circle and the curved surface area of the cylinder. And one more thing to be remembered is that never forget to convert the diameter into radius if diameter is given instead of radius.