(i) The curved surface area of a cylinder is 4400 cm² and the circumference of its base is 110 cm. Find the height and the volume of the cylinder.

(ii) The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. Find the radius of the cylinder and also its volume.

Given, curved surface area of a cylinder = 4400 cm²

(i) By formula,

Curved surface area of cylinder = 2πrh

∴ 2πrh = 4400 ….(1)

Given, circumference of base = 110 cm

We know that circumference = 2πr

∴ 2πr = 110 ….(2)

Dividing eq. (1) by (2),

$\Rightarrow \dfrac{2π\text{rh}}{2π\text{r}} = \dfrac{4400}{110} \\[1em] \Rightarrow \text{h} = 40 \text{ cm.}$

From eq.(1), we have,

$\Rightarrow 2 \times \dfrac{22}{7} \times \text{r} = 110 \\[1em] \Rightarrow \text{r} = \dfrac{110 \times 7}{22 \times 2} \\[1em] \Rightarrow \text{r} = \dfrac{770}{44} \\[1em] \Rightarrow \text{r} = 17.5 \text{ cm.}$

Volume of cylinder = πr²h

Putting values we get,

$= \dfrac{22}{7} \times (17.5)^2 \times 40 \\[1em] = \dfrac{22}{7} \times 306.25 \times 40 \\[1em] = \dfrac{269500}{7} \\[1em] = 38500 \text{ cm}^3.$

Hence, the height of the cylinder is 40 cm and volume of the cylinder = 38500 cm³.

(ii) Given, circumference of base = 132 cm

We know that circumference = 2πr

∴ 2πr = 132

$\Rightarrow 2 \times \dfrac{22}{7} \times \text{r} = 132 \\[1em] \Rightarrow \text{r} = \dfrac{132 \times 7}{22 \times 2} \\[1em] \Rightarrow \text{r} = \dfrac{924}{44} \\[1em] \Rightarrow \text{r} = 21 \text{ cm.}$

Given, height (h) = 25 cm.

Volume of cylinder = πr²h

Putting values we get,

$= \dfrac{22}{7} \times (21)^2 \times 25 \\[1em] = \dfrac{22}{7} \times 441 \times 25 \\[1em] = \dfrac{242550}{7} \\[1em] = 34650 \text{ cm}^3.$

Hence, radius of the cylinder is 21 cm and volume of the cylinder is 34650 cm³.