Question

A solid cylinder has a total surface area of $462c{m^2}$. Its curved surface area is one-third of its total surface area. Find the volume of the cylinder. $\left( {{\text{Take }}\pi = \dfrac{{22}}{7}} \right)$
A) $792c{m^3}$
B) $539c{m^3}$
C) $495c{m^3}$
D) $676c{m^3}$

Answer 1

Hint: In this question, we are given total surface area and curved surface area of a cylinder and we have been asked to find the volume. Using the formula of total surface area, find the value of $2\pi {r^2}$. When you find its value, find the radius of the cylinder. Then put it in the formula of curved surface area to find the height. Now, you have all the specifications, simply put them in the formula of volume and you will have your answer.

Formula used: 1) Total surface area = $2\pi r\left( {r + h} \right)$
2) Curved surface area = $2\pi rh$
3) Volume = $\pi {r^2}h$

Complete step-by-step solution:
We are given the total surface area and the curved surface are of a cylinder and we have not been given any other dimensions. We have been asked to find the volume of the cylinder using the given things.
We know, total surface area = $2\pi r\left( {r + h} \right)$ and it is given in the question to be $462c{m^2}$.
We can also calculate curved surface area as it is given that it is one third of the total surface area.
Therefore, curved surface area = $\dfrac{{462}}{3} = 154c{m^2}$. We also know that curved surface area = $2\pi rh$.
$ \Rightarrow 2\pi r\left( {r + h} \right) = 462c{m^2}$
Opening the brackets,
$ \Rightarrow 2\pi {r^2} + 2\pi rh = 462c{m^2}$
We know curved surface area = $2\pi rh$$ = 154c{m^2}$. Putting in the above equation,
$ \Rightarrow 2\pi {r^2} + 154 = 462$
$ \Rightarrow 2\pi {r^2} = 462 - 154$
$ \Rightarrow 2\pi {r^2} = 308c{m^2}$
Using this equation, we will find the value of r by putting $\pi = \dfrac{{22}}{7}$.
$ \Rightarrow 2 \times \dfrac{{22}}{7} \times {r^2} = 308$
Shifting to find the value of r,
$ \Rightarrow {r^2} = \dfrac{{308 \times 7}}{{2 \times 22}}$
Solving RHS,
$ \Rightarrow {r^2} = 49$
Square rooting both the sides,
$ \Rightarrow r = \pm 7$
Hence, the value or is 7cm.
Now, we will find the value of the height of the cylinder by putting all the values that we have in the formula of curved surface area.
$ \Rightarrow 2\pi rh = 154$
Putting all the values,
$ \Rightarrow 2 \times \dfrac{{22}}{7} \times 7 \times h = 154$
Shifting to find the value of h,
$ \Rightarrow h = \dfrac{{154}}{{22 \times 2}}$
$ \Rightarrow h = \dfrac{7}{2}cm$
Now, we know that Volume = $\pi {r^2}h$.
Putting all the values in the formula,
$ \Rightarrow \pi {r^2}h = \dfrac{{22}}{7} \times {7^2} \times \dfrac{7}{2}$
On solving we will get,
$ \Rightarrow \pi {r^2}h = 539c{m^3}$

$\therefore $ The volume of cylinder = option (B) $539c{m^3}$

Note: We can observe from the solution of the question that the total surface area includes the ends of the cylinders which are circular planes whereas the curved surface area is along the curvature of the cylinder body.
For cylinder: \[{\text{Total surface area = Curved surface area + 2}}\left( {{\text{area of circle}}} \right)\]
This question is relevant to solid cylinders. Curved surface area or lateral surface area is the area of the curved surface on the cylinder. Total surface area is the sum of curved surface area and the flat surface base of a cylinder.
At the step where we calculated the radius of the cylinder, we neglected the value of $r = - 7$.
This is because any radius of any cylinder can never have any negative value. That is why we cancelled out $r = - 7$ and chose $r = 7cm$.