Question

Total surface area of the cylinder is 462 $c{m^2}$. The curved surface area is $\dfrac{1}{3}$ of total surface area. Find volume.

Answer 1

Hint: Total surface area is given by $2\pi r(h + r)$. Curved surface area (CSA) is the area of a curved surface of a cylinder. Curved surface area is given by $2\pi rh$. Volume of the cylinder is given by $\pi {r^2}h$. In this question “r” is the radius of the cylinder and “h” is the height of the cylinder.

Complete step by step answer:
In this question, the total surface area of the cylinder is given and curved surface area is one third of total surface area. To find volume we need , radius and height of the cylinder. First let’s find the radius and height of the cylinder with the help of a statement given that its curved surface area is one third of total surface area.
Total surface area $ = 462c{m^2}$
We know the formula of total surface area
$2\pi r(h + r) = 462c{m^2}$
It can be written as
$2\pi rh + 2\pi {r^2} = 462c{m^2}$ ……………..(1)
Curved surface area $ = \dfrac{1}{3} \times $ Total surface area
Curved surface area $ = \dfrac{1}{3} \times $$462$
Solving above , we get
$ \Rightarrow $Curved surface = $154c{m^2}$
We know the formula of curved surface area is $2\pi rh$
$ \Rightarrow $$2\pi rh$= $154c{m^2}$ ………………………(2)
Putting the value of $2\pi rh$ in equation 1
$2\pi rh + 2\pi {r^2} = 462c{m^2}$
$ \Rightarrow 154 + 2\pi {r^2} = 462$
$
   \Rightarrow 2\pi {r^2} = 462 - 154 \\
   \Rightarrow 2\pi {r^2} = 308c{m^2} \\
 $
We know $\pi = \dfrac{{22}}{7}$
$
   \Rightarrow 2 \times \dfrac{{22}}{7}{r^2} = 308 \\
   \Rightarrow {r^2} = 49 \\
 $
Square root of 49 is 7
$
   \Rightarrow r = \sqrt {49} \\
   \Rightarrow r = 7cm \\
 $
Putting the value of “r” in equation 2
$2\pi rh$= $154c{m^2}$
We know $\pi = \dfrac{{22}}{7}$
$
   \Rightarrow 2 \times \dfrac{{22}}{7} \times 7 \times h = 154 \\
   \Rightarrow h = 3.5cm \\
 $
Now, putting the value of “r” , “ h” and $\pi $ in formula of volume of cylinder
Volume of cylinder = $\pi {r^2}h$
$ \Rightarrow \dfrac{{22}}{7} \times 7 \times 7 \times 3.5$
$ \Rightarrow 539c{m^3}$

Hence, the volume of the cylinder is $539c{m^3}$.

Note:
Curved surface area is also known as lateral surface area (LSA).
Total surface area (TSA) is the sum of curve area and circular areas of cylinder.