Question

The total surface area of the hollow cylinder, which is open from both sides is $3575c{m^2}$; area of the base ring is $357.5c{m^2}$ and height is $14cm$. Find the thickness of the cylinder.

Answer 1

Hint: The total surface area of the hollow cylinder (pipe) is given which is the sum of inner curved surface area, outer curved surface area and twice the area of base ring. First of all, suppose the inner radius as $r$ and outer radius as $R$ and then write the area of the base ring and equate it with the given area. Then write inner and outer curved surface area and total surface area and equate it with the given total surface area. Finally, the thickness of the cylinder is calculated by subtracting the inner radius from the outer radius.

Complete answer:
Given, the total surface area is $3575c{m^2}$.
Area of the base ring is $357.5c{m^2}$ and height of the cylinder is $14cm$.
Let the inner radius of the cylinder be $r$and the outer radius be \[R\] $cm$.
We know that the area of the circle is $\pi {r^2}$ and the curved surface area of the cylinder $ = 2\pi rh$.
Area of the base ring is the difference of the area of the outer circle of radius $R$ and the area of inner circle of radius $r$. So,
Area of the base ring $ = \pi \left( {{R^2} - {r^2}} \right)$
Now, equating it with the given area of the base ring. We get,
$ \Rightarrow \pi \left( {{R^2} - {r^2}} \right) = 357.5$
By using a mathematical identity $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$. We can write
$ \Rightarrow \pi \left( {R + r} \right)\left( {R - r} \right) = 357.5$-----------------(1)
Now, inner curved surface area of the hollow cylinder$ = 2\pi rh$
Outer curved surface area of the hollow cylinder $ = 2\pi Rh$
The total surface area $ = 2\pi Rh + 2\pi rh + 2 \times \pi \left( {{R^2} - {r^2}} \right)$
Put $\pi \left( {{R^2} - {r^2}} \right) = 357.5$ and then equating the total surface area with the given total surface area we get,
$
   \Rightarrow 2\pi Rh + 2\pi rh + 2 \times 357.5 = 3575 \\
   \Rightarrow 2\pi h\left( {R + r} \right) + 715 = 3575 \\
$
Putting the value of $\pi = \dfrac{{22}}{7}$ and $h = 14cm$. We get,
$
   \Rightarrow 2 \times \dfrac{{22}}{7} \times 14 \times \left( {R + r} \right) = 3575 - 715 \\
   \Rightarrow 2 \times 22 \times 2 \times \left( {R + r} \right) = 2860 \\
   \Rightarrow 88 \times \left( {R + r} \right) = 2860 \\
   \Rightarrow \left( {R + r} \right) = \dfrac{{2860}}{{88}} \\
   \Rightarrow \left( {R + r} \right) = \dfrac{{65}}{2} \\
  \therefore \left( {R + r} \right) = 32.5 \\
 $
Now, putting the value of $\left( {R + r} \right)$ in equation (1), we get,
$
   \Rightarrow \pi \times 32.5\left( {R - r} \right) = 357.5 \\
   \Rightarrow \dfrac{{22}}{7} \times 32.5 \times \left( {R - r} \right) = 357.5 \\
   \Rightarrow \left( {R - r} \right) = \dfrac{{357.5 \times 7}}{{22 \times 32.5}} \\
  \therefore \left( {R - r} \right) = 3.5 \\
$
The thickness of the cylinder is the difference of the outer radius and inner radius i.e $\left( {R - r} \right)$.

Hence, the thickness of the cylinder is $3.5cm$.

Note: Similarly, we can calculate the volume of the material in the hollow cylinder. We have to find the volume of outer cylinder by using the formula $\pi {R^2}h$ and then the volume of inner cylinder then the difference of outer and inner volume of the cylinder gives the volume of materials in the hollow cylinder.