Question

The curved surface area of a cylinder is 264 m2 and

its volume is 924 m3 . Find the ratio of its height to its

diameter.

Answer 1

Dear Student,
Solution)
​​​​​​Suppose r and h be the radius and the height of the cylinder respectively.
So, curved surface area of cylinder = $2\mathrm{\pi rh}=264{\mathrm{m}}^2$
$$\begin{gathered} \Rightarrow 2rh=\frac{264}{\mathrm{\pi }} \\ \Rightarrow 2rh=\frac{264\times 7}{22}=84.....\left(\mathrm{i}\right) \end{gathered}$$

and
Volume of cylinder = ${\mathrm{\pi r}}^2\mathrm{h}=924{\mathrm{m}}^3$
$$\begin{gathered} \Rightarrow {\mathrm{r}}^2\mathrm{h}=\frac{924}{\mathrm{\pi }} \\ \Rightarrow {\mathrm{r}}^2\mathrm{h}=\frac{924\times 7}{22}=294....\left(\mathrm{ii}\right) \end{gathered}$$
Now, on dividing (ii) by (i), we get,
$$\begin{gathered} \frac{r^{2}h}{2rh}=\frac{294}{84} \\ \Rightarrow \frac{r}{2}=\frac{294}{84} \\ \Rightarrow r=\frac{294}{84}\times 2=7 \end{gathered}$$
Therefore, radius of cylinder = 7 m
Then its diameter = $2\times 7$ = 14 m
{because diameter = 2 × radius}
Putting the value of r in (i), we get,
$$\begin{gathered} 2\times 7h=84 \\ \Rightarrow h=\frac{84}{2\times 7}=6 \end{gathered}$$
So height of cylinder is 6 m.
so the ratio of height to the diameter = $\frac{6}{14}=\frac{3}{7}$
Thus, the ratio of height to the diameter is 3 : 7.

Regards!