A hollow cylinder of height h has inner and outer radii of base given as r and R, respectively. The ratio of the curved surface area to the total surface area of this cylinder is:

Correct option is A

​Given:

Height: h.

Inner radius: r.

Outer radius: R.

Formulas Used:

Curved surface area (CSA): $$\text{CSA} = 2\pi h (R + r)$$​​

Total surface area (TSA): $$\text{TSA} = 2\pi h (R + r) + 2\pi (R^2 - r^2)$$​​

Solution:
$$\text{Ratio} = \frac{CSA}{TSA}$$​​

​$$\text{Ratio} = \frac{2\pi h (R + r)}{2\pi h (R + r) + 2\pi (R^2 - r^2)}$$​​

$$\text{Ratio} = \frac{h (R + r)}{h (R + r) + (R^2 - r^2)}$$

​$$\text{Ratio} = \frac{h}{h + \frac{R^2 - r^2}{R + r}}$$​​

​$$[R^2 - r^2 = (R - r)(R + r)]$$​

​$$\text{Ratio} = \frac{h}{h + \frac{(R - r)(R + r)}{R + r}}$$​

​$$Ratio=h :h+(R−r)$$​​