Question

The ratio of radii of two right circular cylinders is $2:3$ and the ratio of their heights is $5:4$. Find the ratios of their curved surface areas and volumes.

Answer 1

Let the radius of first cylinder be $r_1$ and height be $h_1$ and let the radius of second cylinder be $r_2$ and height be $h_2.$
Then according to question $\frac{r_1}{r_2}=\frac{2}{3}$
and $\frac{h_1}{h_2}=\frac{5}{4}$
Curved surface area of first cylinder $S_1=2\pi r_1{h}_1$
and, curved surface area of second cylinder $S_2=2\pi r_2{h}_2$
$\begin{array}{cc}\therefore \frac{S_1}{S_2}& =\frac{2\pi r_1{h}_1}{2\pi r_2{h}_2}\\ & =\left(\frac{r_1}{r_2}\right)\left(\frac{h_1}{h_2}\right)\\ & =\left(\frac{2}{3}\right)\left(\frac{5}{4}\right)\\ & =\frac{5}{6}\\ & S_1:S_2=5:6\end{array}$
The ratio of their volume
$\begin{array}{cc}\frac{V_1}{V_2}& =\frac{\pi r_1^2{h}_1}{\pi r_2^2{h}_2}\\ & ={\left(\frac{2}{3}\right)}^2\cdot \left(\frac{5}{4}\right)\\ & =\frac{4}{9}\times \frac{5}{4}\\ & =\frac{5}{9}\\ \therefore V_1:V_2& =5:9\end{array}$
Hence, ratio of curved surface areas $=5:6$
and ratio of volumes $=5:9$