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 $616c{m}^{2.}$

Answer 1

Step 1: Find the radius and height of the cylinder.

Let, $r,h$ be the radius and height of the cylinder respectively.

Given that, $\frac{curvedsurfaceareaofcylinder}{totalsurfaceareaofcylinder}=\frac{1}{2}$, $totalsurfacearea=616c{m}^2$

The curved surface area of cylinder of radius $r$ and height $h$ $=2\mathrm{\pi rh}$

The total surface area of a cylinder of radius $r$ and height $h$ $=2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{h}\right)$

Given that, $\frac{curvedsurfaceareaofcylinder}{totalsurfaceareaofcylinder}=\frac{1}{2}$

$$\begin{gathered} \therefore \frac{2\mathrm{\pi r}h}{2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{h}\right)}=\frac{1}{2} \\ \Rightarrow 2h=r+h \\ \Rightarrow r=h \end{gathered}$$

Now, we know $2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{h}\right)=616{\mathrm{cm}}^2$

$$\begin{gathered} \Rightarrow 2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{r}\right)=616{\mathrm{cm}}^2 \\ \Rightarrow 4{\mathrm{\pi r}}^2=616{\mathrm{cm}}^2 \\ \Rightarrow {\mathrm{r}}^2=49{\mathrm{cm}}^2 \\ \Rightarrow \mathrm{r}=7\mathrm{cm} \end{gathered}$$

So, $r=h=7cm$

Step 2: Find the volume of cylinder:

The volume of cylinder of radius $r$ and height $h$ $={\mathrm{\pi r}}^2\mathrm{h}$

$$\begin{gathered} ={\mathrm{\pi r}}^3 \\ =\frac{22}{7}\times {\left(7\right)}^3 \\ =1078{\mathrm{cm}}^3 \end{gathered}$$

Therefore, volume of cylinder is $1078c{m}^3$.