Question

The capacity of a closed cylindrical vessel of height 1 m is 15.4 liters. How many square meters of metal sheet would be needed to make it?

Answer 1

Hint: In order to solve this question we will equate the capacity of a closed cylindrical vessel to the volume of cylinder so that we will get the value of radius and height value is given so we will proceed further by applying the formula of surface area of cylinder which is mentioned in the solution.

Complete step-by-step answer:
We will use the following figure to solve the problem.

Given that height of cylindrical vessel is 1m
And capacity of closed cylindrical vessel is 15.4 liters
As we know that the volume of the cylinder will be calculated in meter units, first we will convert this liter unit to the \[{m^3}\] unit.
As we know the conversion rule is:
$
  1{m^3} = 1000liters \\
   \Rightarrow \dfrac{1}{{1000}}{m^3} = 1liter \\
   \Rightarrow 1liter = \dfrac{1}{{1000}}{m^3} \\
 $
Capacity of cylindrical vessel = Volume of cylinder
$
   = 15.4liters \\
   = 15.4 \times \dfrac{1}{{1000}}{m^3} \\
   = 154 \times \dfrac{1}{{10000}}{m^3} \\
   = \dfrac{{154}}{{10000}}{m^3} \\
 $
We know that volume of cylinder is given as
$v = \pi {r^2}h$
Substitute the value of volume and height in above formula, we have
$
  \because v = \pi {r^2}h \\
   \Rightarrow \dfrac{{154}}{{1000}} = \dfrac{{22}}{7}{r^2}1 \\
 $
Let us find the value of r by solving the equation:
\[
   \Rightarrow \dfrac{{22}}{7}{r^2} = \dfrac{{154}}{{10000}} \\
   \Rightarrow {r^2} = \dfrac{{154}}{{10000}} \times \dfrac{7}{{22}} \\
   \Rightarrow {r^2} = \dfrac{7}{{10000}} \times \dfrac{7}{1} \\
   \Rightarrow {r^2} = \dfrac{{{7^2}}}{{{{100}^2}}} \\
 \]
We will further simplify it by taking root to both the sides, we obtain
\[
   \Rightarrow r = \sqrt {\dfrac{{{7^2}}}{{{{100}^2}}}} \\
   \Rightarrow r = \dfrac{7}{{100}} \\
   \Rightarrow r = 0.07m \\
 \]
We need to find square meters of metal sheet needed to make it, i.e. total surface area of the cylinder as it is a closed vessel.
We know that total surface area of cylinder $ = 2\pi r\left( {r + h} \right)$
Substitute the value of radius and height, we have:
\[
  \because A = 2\pi r\left( {r + h} \right) \\
   \Rightarrow A = 2 \times \dfrac{{22}}{7} \times \dfrac{7}{{100}}\left( {\dfrac{7}{{100}} + 1} \right) \\
   \Rightarrow A = 2 \times 22 \times \dfrac{1}{{100}}\left( {\dfrac{{7 + 100}}{{100}}} \right) \\
   \Rightarrow A = 44 \times \dfrac{1}{{100}}\left( {\dfrac{{107}}{{100}}} \right) \\
   \Rightarrow A = \dfrac{{4708}}{{10000}} \\
   \Rightarrow A = 0.4708{m^2} \\
 \]
Hence, the metal sheet needed to make a closed cylindrical vessel is \[0.4708{m^2}\].

Note- In order to solve such a problem where one of the values of the shape is given along with some data. First consider the unknown value of the side of the shape as some unknown and use the data to find it. Further the problem will become easier to find any other quantity for the figure. Students must remember the relation between the volume in liter and cubic meter and also the formulas for areas and volume of shapes.