Question

A solid cylinder has a total surface area of \[462{\text{ }}c{m^2}\] . Its curved surface area is one-third of its total surface area. Find the volume of the cylinder

Answer 1

Hint: To find the volume of the solid cylinder need to know the value of radius of the cylinder. So to get the value of the radius use the concept that the total surface area of the solid cylinder is the sum of the curved surface area and two times the area of the circle of the cylinder. Once the radius is found use the formula of volume of the solid cylinder by putting the value of the radius.

Complete step-by-step answer:
Given, total surface area of cylinder = \[462{\text{ }}c{m^2}\] .
Also, \[\;CSA\] of cylinder \[ = \dfrac{{TSA}}{3}\;\] of cylinder​
Where \[\;CSA\] is the curved surface area of the cylinder and \[TSA\] is the total surface area of the cylinder
We know the total surface area of the solid cylinder is the sum of the curved surface area and two times the area of the circle
 \[ \Rightarrow TSA = 2\pi rh + 2\pi {r^2}\]
Or,
 \[ \Rightarrow TSA = CSA + 2\pi {r^2}\]
Or,
 \[ \Rightarrow TSA = \dfrac{{TSA}}{3} + 2\pi {r^2}\]
Or,
 \[ \Rightarrow \dfrac{{2TSA}}{3} = 2\pi {r^2}\]
Or,
 \[ \Rightarrow \dfrac{{TSA}}{3} = \pi {r^2}\]
On putting the value of the total surface area
 \[ \Rightarrow \dfrac{{462}}{3} = \dfrac{{22}}{7} \times {r^2}\]
We get the value of the radius
 \[ \Rightarrow r = 7cm\;\]
Now, CSA of cylinder \[ = \dfrac{{TSA\;}}{3}\] of cylinder \[ = \dfrac{{462}}{3} = 154c{m^2}\]
On equating the value of the CSA
We get,
 \[\begin{array}{*{20}{l}}
  { \Rightarrow 2\pi rh = 154} \\
  { \Rightarrow 2 \times \dfrac{{22}}{7} \times 7 \times h = 154} \\
  { \Rightarrow h = 27cm}
\end{array}\]
Therefore, volume of the cylinder \[ = \pi {r^2}h\]
On putting the value of the height and radius
We get the volume of the solid cylinder
 \[ = \dfrac{{22}}{7} \times {7^2} \times 27 = 539{\text{ }}c{m^3}\]
Hence, the volume of the solid cylinder is \[539{\text{ }}c{m^3}\] .
So, the correct answer is “ \[539{\text{ }}c{m^3}\] ”.

Note: Here in this question there are two variable radius and height of the solid cylinder at once both the variables can not be found so find the radius first by using the value of the total surface then secondly find the value of height by using the value of curved surface area.