A cylinder of circumference 8 cm and length 21 cm rolls without sliding for `4 (1)/(2) ` seconds at the rate of 9 complete rounds per second. Find :
the area covered by the cylinder in `4 (1)/(2)` seconds.

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Circumference of the cylinder (C) = 8 cm
- Length (height) of the cylinder (h) = 21 cm
- Time (t) = 4.5 seconds
- Rate of revolutions = 9 revolutions per second

2. **Calculate the Radius of the Cylinder:**
The formula for the circumference of a circle is given by:
\[ C = 2 \pi r \]
Rearranging this to find the radius (r):
\[ r = \frac{C}{2 \pi} \]
Substituting the value of C:
\[ r = \frac{8}{2 \times \frac{22}{7}} = \frac{8 \times 7}{2 \times 22} = \frac{56}{44} = \frac{14}{11} \text{ cm} \]

3. **Calculate the Curved Surface Area (CSA) of the Cylinder:**
The formula for the curved surface area of a cylinder is:
\[ \text{CSA} = 2 \pi r h \]
Substituting the values of r and h:
\[ \text{CSA} = 2 \times \frac{22}{7} \times \frac{14}{11} \times 21 \]
Simplifying this:
\[ = \frac{44}{7} \times \frac{14 \times 21}{11} \]
\[ = \frac{44 \times 14 \times 21}{77} \]
\[ = \frac{44 \times 14 \times 3}{11} = 8 \times 21 = 168 \text{ cm}^2 \]
This is the area covered in one complete revolution.

4. **Calculate the Total Area Covered in 4.5 Seconds:**
Since the cylinder rolls at a rate of 9 revolutions per second, in 4.5 seconds it will complete:
\[ \text{Total revolutions} = 9 \times 4.5 = 40.5 \text{ revolutions} \]
Therefore, the total area covered is:
\[ \text{Total Area} = \text{Area per revolution} \times \text{Total revolutions} \]
\[ = 168 \times 40.5 = 6804 \text{ cm}^2 \]

### Final Answer:
The area covered by the cylinder in 4.5 seconds is **6804 cm²**.