Question

The ratio between the curved surface area and the total surface area of a right circular cylinder is 1: 2. Find the volume of the cylinder if its total surface area is 616 cm sq.

Answer 1

Hint: In this question directly use the formula for the total surface area of a right circular cone and the curved surface area of the right circular cone as it helps in getting the relationship between height and radius of base and eventually their values.

Complete step-by-step answer:

As we know that the curved surface area (C.S.A) of the right circular cylinder is $2\pi rh$.
And the total surface area (T.S.A) of the right circular cylinder is $\left( {2\pi rh + 2\pi {r^2}} \right)$.
Where r = base radius of the cylinder, h = height of the cylinder.
Now it is given that the ratio of curved surface area and the total surface area is (1: 2).
$ \Rightarrow \dfrac{{C.S.A}}{{T.S.A}} = \dfrac{1}{2}$
$ \Rightarrow \dfrac{{2\pi rh}}{{2\pi rh + 2\pi {r^2}}} = \dfrac{1}{2}$
Now simplify the above equation we have,
$ \Rightarrow 4\pi rh = 2\pi rh + 2\pi {r^2}$
$ \Rightarrow 2\pi rh = 2\pi {r^2}$
Divide by $2\pi r$we have,
$ \Rightarrow h = r$……………………… (1)
Now it is given that the total surface area is 616 cm sq.
$ \Rightarrow 2\pi rh + 2\pi {r^2} = 616{\text{ c}}{{\text{m}}^2}$
From equation (1) we have
$
   \Rightarrow 2\pi {r^2} + 2\pi {r^2} = 616 \\
   \Rightarrow 4\pi {r^2} = 616 \\
 $
$ \Rightarrow {r^2} = \dfrac{{616}}{4} \times \dfrac{7}{{22}} = 49 = {7^2}$ $\left[ {\because \pi = \dfrac{{22}}{7}} \right]$
$ \Rightarrow r = 7$ cm
Now we know that the volume (V) of cylinder is $ = \pi {r^2}h$
$ \Rightarrow V = \dfrac{{22}}{7} \times {\left( 7 \right)^2} \times 7 = 22 \times 49 = 1078{\text{ c}}{{\text{m}}^3}$
So this is the required volume of the right circular cylinder.
So this is the required answer.

Note: Whenever we face such types of problems the key concept is simply to have a good grasp over the basic formula for certain conic sections like right circular cone, cylinder, hemisphere etc. This helps save a lot of time as the information provided in the question can directly be used for formulation of equations.