A cylinder of circumference 8 cm and length 21 cm rolls without sliding for $4\dfrac{1}{2}$ seconds at the rate of 9 complete rounds per second. Find:

(i) distance travelled by the cylinder in $4\dfrac{1}{2}$ seconds, and

(ii) the area covered by the cylinder in $4\dfrac{1}{2}$ seconds

(i) If distance covered in one revolution is 8 cm, then distance covered in 9 revolutions = 9 x 8 = 72 cm or distance covered in 1 second = 72 cm.

Distance covered in $4\dfrac{1}{2}$ seconds = 72 x 4.5 = 324 cm.

Hence, distance travelled by the cylinder in $4\dfrac{1}{2}$ seconds is 324 cm.

(ii) Given, circumference = 8 cm.

∴ 2πr = 8

⇒ $2 \times \dfrac{22}{7} \times r = 8$

⇒ r = $\dfrac{8 \times 7}{2 \times 22} = \dfrac{14}{11}$ cm.

By formula,

Curved surface area = 2πrh

= 2 x $\dfrac{22}{7} \times \dfrac{14}{11}$ x 21

= 2 x 2 x 2 x 21

= 168 cm².

So, the area covered in one revolution = 168 cm²,

∴ The area covered in 9 revolutions = 168 x 9 = 1512 cm²,

∴ The area covered in 1 second = 1512 cm²,

Thus, the area covered in $4\dfrac{1}{2}$ seconds = 1512 x 4.5 = 6804 cm².

Hence, the area covered by cylinder in $4\dfrac{1}{2}$ seconds = 6804 cm².