In a right circular cylinder, the ratio of the curved surface area to the total surface area is 5 : 9. Find the ratio of the height of the cylinder to the radius of the cylinder.

Ratio of the curved surface area (CSA) to the total surface area (TSA) of the cylinder is 5:9.

Curved Surface Area (CSA) of a cylinder:

​$$\text{CSA} = 2\pi r h$$​​

Total Surface Area (TSA) of a cylinder:

​$$\text{TSA} = 2\pi r h + 2\pi r^2$$​​

Let the height of the cylinder be h and the radius be r.

Given that the ratio of CSA to TSA is 5:9,

So,

​$$\frac{2\pi r h}{2\pi r h + 2\pi r^2} = \frac{5}{9}$$ ​

​$$\frac{h}{h + r} = \frac{5}{9}$$​​

9h = 5(h + r)

9h = 5h + 5r

9h - 5h = 5r

4h = 5r

Thus, the ratio of the height to the radius is:

​$$\frac{h}{r} = \frac{5}{4}$$​​