Question:
Find the ratio between the total surface area of a cylinder to its curved surface area, given that its height and radius are 7.5 cm and 3.5 cm.
Solution:
Let $S_{1}$ and $S_{2}$ be the total surface area and curved surface area, respectively.
Given :
Height, $h=7.5 \mathrm{~cm}$
Radius, $r=3.5 \mathrm{~cm}$
$\mathrm{~S}_{1}=2 \pi r(r+h)$
$\mathrm{S}_{2}=2 \pi r h$
According to the question:
$\frac{S_{1}}{S_{2}}=\frac{2 \pi r(r+h)}{2 \pi r h}$
$\frac{S_{1}}{S_{2}}=\frac{\mathrm{r}+\mathrm{h}}{\mathrm{h}}$
$\frac{S_{1}}{S_{2}}=\frac{3.5+7.5}{7.5}$
$\frac{S_{1}}{S_{2}}=\frac{11}{7.5}=\frac{110}{75}=\frac{22}{15}$
Therefore, the ratio is $22: 15$.