Question

The radius of the base and the height of a solid right circular cylinder are in the ratio 2 : 3 and its volume is 1617 cm³. Find the total surface area of the cylinder.

Answer 1

Let the radius of the cylinder be 2x cm and its height be 3x cm.

Then, volume of the cylinder$=\mathrm{\pi }{r}^{2}h$
$=\frac{22}{7}\times {\left(2x\right)}^2\times 3x$
Therefore,
$$\begin{gathered} \frac{22}{7}\times {\left(2x\right)}^2\times 3x=1617 \\ \Rightarrow \frac{22}{7}\times 4x^2\times 3x=1617 \\ \Rightarrow \frac{22}{7}\times 12x^3=1617 \\ \Rightarrow x^3=\left(1617\times \frac{7}{22}\times 12\right) \end{gathered}$$

$$\begin{gathered} \Rightarrow x^3=\left(\frac{7}{2}\times \frac{7}{2}\times \frac{7}{2}\right) \\ \Rightarrow x^3={\left(\frac{7}{2}\right)}^3 \\ \Rightarrow x=\frac{7}{2} \end{gathered}$$
Now, $r=7\text{cm and}h=\frac{21}{2}\text{cm}$
Hence, the total surface area of the cylinder:

$\left(2\mathrm{\pi }rh+2\mathrm{\pi }{r}^2\right)$
​ $$\begin{gathered} =2\mathrm{\pi }r\left(h+r\right) \\ =2\times \frac{22}{7}\times 7\times \left(\frac{21}{2}+7\right){\text{cm}}^2 \\ =\left(2\times \frac{22}{7}\times 7\times \frac{35}{2}\right){\text{cm}}^2 \\ =770{\text{cm}}^2 \end{gathered}$$