Question

The ratio between the radius of the base and the height of a cylinder is 2 : 3. If its volume is 1617 cm³, the total surface area of the cylinder is
(a) 308 cm²
(b) 462 cm²
(c) 540 cm²
(d) 770 cm²

Answer 1

(d) 770 cm²
Let the common multiple be x.
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,
$$\begin{gathered} \frac{22}{7}\times {\left(2x\right)}^2\times 3x=1617 \\ \Rightarrow \frac{22}{7}\times 4x^2\times 3x=1617 \\ \Rightarrow \frac{22}{7}\times 12x^3=1617 \\ \Rightarrow x^3=\left(1617\times \frac{7}{22}\times \frac{1}{12}\right) \end{gathered}$$

$$\begin{gathered} \Rightarrow x^3=\left(\frac{7}{2}\times \frac{7}{2}\times \frac{7}{2}\right) \\ \Rightarrow x^3={\left(\frac{7}{2}\right)}^3 \\ \Rightarrow x=\frac{7}{2} \end{gathered}$$
Now, $r=7\text{cm and}h=\frac{21}{2}\text{cm}$
Hence, the total surface area of the cylinder$=\left(2\mathrm{\pi }rh+2\mathrm{\pi }{r}^2\right)$
$$\begin{gathered} =2\mathrm{\pi }r\left(h+r\right) \\ =2\times \frac{22}{7}\times 7\times \left(\frac{21}{2}+7\right){\text{cm}}^2 \\ =\left(2\times \frac{22}{7}\times 7\times \frac{35}{2}\right){\text{cm}}^2 \\ =770{\text{cm}}^2 \end{gathered}$$