Question

The radius of the base and the height of a right circular cylinder are in the ratio of: $2:3$ and its volume is $1617cu.cm$. Find the total surface area of the cylinder. (Use $\mathrm{\pi }=\frac{22}{7}$​)

Answer 1

Step 1: Determine the value of radius and height

The ratio of the radius and height of the cylinder is $2:3$

$$\begin{gathered} \frac{r}{h}=\frac{2}{3} \\ \Rightarrow r=\frac{2}{3}h \end{gathered}$$

The volume of the cylinder is $1617c{m}^3$

$$\begin{gathered} {\mathrm{\pi r}}^2\mathrm{h}=1617 \\ \Rightarrow \mathrm{\pi }\times {\left(\frac{2}{3}\mathrm{h}\right)}^2\times \mathrm{h}=1617 \\ \Rightarrow \frac{4}{9}{\mathrm{\pi h}}^3=1617 \\ \Rightarrow {\mathrm{h}}^3=\frac{1617}{4\mathrm{\pi }}\times 9 \\ \Rightarrow {\mathrm{h}}^3=\frac{1617}{4\times {\displaystyle \frac{22}{7}}}\times 9 \\ \Rightarrow {\mathrm{h}}^3=\frac{1617\times 9\times 7}{4\times 22} \\ \Rightarrow {\mathrm{h}}^3=\frac{147\times 9\times 7}{8} \\ \Rightarrow {\mathrm{h}}^3=\frac{3^3\times 7^3}{8} \\ \Rightarrow {\mathrm{h}}^3={\left(\frac{21}{8}\right)}^3\Rightarrow \mathrm{h}=\frac{21}{2}\mathrm{cm} \end{gathered}$$

$$\begin{gathered} r=\frac{2}{3}\times h \\ =\frac{2}{3}\times \frac{21}{2} \\ =7cm \end{gathered}$$

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

The total surface area of the cylinder:

$$\begin{gathered} =2\mathrm{\pi r}(\mathrm{r}+\mathrm{h}) \\ =2\times \frac{22}{7}\times 7(7+\frac{21}{2}) \\ =44\times \left(\frac{14+21}{2}\right) \\ =22\times 35 \\ =770{\mathrm{cm}}^2 \end{gathered}$$

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