The radius and height of a cylinder are in the ratio 5 : 7

$\frac{\text { Radius }}{\text { height }}=\frac{r}{h}=\frac{5}{7}$

$\Rightarrow r=\frac{5}{7} h$

Now, volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times \frac{5}{7} \mathrm{~h} \times \frac{5}{7} \mathrm{~h} \times \mathrm{h}=550 \mathrm{~cm}^{3}$

$\therefore h=\sqrt[3]{\frac{550 \times 7 \times 7 \times 7}{22 \times 5 \times 5}}=7 \mathrm{~cm}$

Also, $r=\frac{5}{7} h=5 \mathrm{~cm}$