Question

The ratio between the radius of the base and the height of a cylinder is $2:3$. Find the total surface area of the cylinder, if its volume is $1617c{m}^3$ .

Answer 1

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

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

Given that, $\frac{r}{h}=\frac{2}{3}$, $volume=1617c{m}^3$

$\Rightarrow r=2x,h=3x$

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

$$\begin{gathered} \therefore {\mathrm{\pi r}}^2\mathrm{h}=1617{\mathrm{cm}}^3 \\ \Rightarrow \mathrm{\pi }{\left(2\mathrm{x}\right)}^2\left(3\mathrm{x}\right)=1617{\mathrm{cm}}^3 \\ \Rightarrow 12{\mathrm{\pi x}}^3=1617{\mathrm{cm}}^3 \\ \Rightarrow {\mathrm{x}}^3=42.875{\mathrm{cm}}^3 \\ \Rightarrow \mathrm{x}=3.5\mathrm{cm} \end{gathered}$$

So, $r=2x=7cm$

And, $h=3x=10.5cm$

Step 2: Find the total surface area of the cylinder:

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

$$\begin{gathered} =2\mathrm{\pi }\left(7\right)\left(7+10.5\right) \\ =770{\mathrm{cm}}^2 \end{gathered}$$

Hence, the total surface area of the cylinder is $770c{m}^{2.}$