Question

The hollow metal pipe is 63 cm long. The inner diameter of cross-section is 4 cm and the outer diameter is 4.8 cm. Find the total surface area of the pipe.

Answer 1

Hint: The total surface area of a hollow pipe is found by adding the curved surface areas of the outer and inner cylindrical parts and the areas of the circular bases where the area of each base is the difference between the area of the outer circle and the area of the inner circle.

Complete step-by-step answer:
We are given the dimensions of a hollow metal pipe.
The length is given as 63 cm. As the pipe is cylindrical in shape, we can say that its height is 63 cm.
We are also given the inner and outer diameters of the cross-section which are 4 cm and 4.8 cm respectively.
We are asked to calculate the total surface area of the pipe.
Let us denote the length by the letter h. Then we have h = 63 cm.
Now, the inner radius of the cross-section = inner diameter \[ \div 2 = 4 \div 2 = 2\]
Let us denote the inner radius by $r$.
Then we have $r = 2cm$.
Similarly, let us denote the outer radius by $R$.
Then $R = 4.8 \div 2 = 2.4cm$

The total surface area of a hollow pipe is given by the formula:
Curved surface area of the inner cylindrical part $ + $ curved surface area of the outer cylindrical part + area of the two circular ends or bases.
We know that the area of each circular base for a hollow pipe
$ = $ area of the outer circle $ - $ area of the inner circle
$
   = \pi {R^2} - \pi {r^2} \\
   = \pi ({R^2} - {r^2}) \\
 $
Now, we substitute the values of R and r, to get
area of each circular base
 $
   = \pi ({(2.4)^2} - {2^2}) \\
   = 1.76\pi c{m^2} \\
 $
As area of 1 circular base is $1.76\pi c{m^2}$, area of two circular bases will be $2 \times 1.76\pi c{m^2} = 3.52\pi c{m^2}....(1)$
Curved surface area of a cylinder is given by the formula
Curved surface area$ = 2\pi \times radius \times height$
Therefore, by substituting the values of radius and height, we get
Curved surface area of outer cylinder$ = 2\pi Rh = 2 \times \pi \times 2.4 \times 63 = 302.4\pi c{m^2}.....(2)$.
Similarly, we get curved surface area of inner cylinder$ = 2\pi rh = 2 \times \pi \times 2 \times 63 = 252\pi c{m^2}....(3)$
We know that the total surface area of a hollow pipe is given by the formula:
Total surface area = Curved surface area of the inner cylindrical part $ + $ curved surface area of the outer cylindrical part + area of the two circular ends or bases.
On substituting the values of (1), (2), and (3), we get
Total surface area
$
  = 252\pi + 302.4\pi + 3.52\pi \\
  = (252 + 302.4 + 3.52)\pi \\
  = 557.92\pi \\
 $
Taking the value of $\pi $as $\dfrac{{22}}{7}$, we get the total surface area $ = 557.92 \times \dfrac{{22}}{7} = 1753.46c{m^2}$
Hence the required total surface area of the hollow pipe is $1753.46c{m^2}$.

Note: It is important to note that total surface area of a hollow pipe cannot be found by just using the formula of total surface area of the cylinder. The question mentioning the inner and outer diameters can help you remember this point.