50 identical circular plates are placed one above the other to form a solid cylinder. For each plate, diameter = 21 cm and thickness = 15 cm. Find :
(i) curved surface area of the cylinder formed.
(ii) volume of the cylinder fomred.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the radius of the cylinder
Given the diameter of the circular plates is 21 cm, we can find the radius (r) using the formula:
\[ r = \frac{\text{diameter}}{2} = \frac{21 \text{ cm}}{2} = 10.5 \text{ cm} \]

### Step 2: Calculate the height of the cylinder
The height (h) of the cylinder is equal to the thickness of each plate multiplied by the number of plates. Given that the thickness of each plate is 15 cm and there are 50 plates:
\[ h = \text{thickness} \times \text{number of plates} = 15 \text{ cm} \times 50 = 750 \text{ cm} \]

### Step 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 \(h\):
\[ \text{CSA} = 2 \times \frac{22}{7} \times 10.5 \text{ cm} \times 750 \text{ cm} \]
Calculating this step-by-step:
1. Calculate \(2 \times \frac{22}{7} = \frac{44}{7}\)
2. Calculate \(10.5 \times 750 = 7875\)
3. Now multiply:
\[ \text{CSA} = \frac{44}{7} \times 7875 \]
4. Dividing \(7875\) by \(7\) gives approximately \(1125\).
5. Finally, calculate \(44 \times 1125 = 49500 \text{ cm}^2\).

### Step 4: Calculate the volume of the cylinder
The formula for the volume (V) of a cylinder is:
\[ V = \pi r^2 h \]
Substituting the values of \(r\) and \(h\):
\[ V = \frac{22}{7} \times (10.5 \text{ cm})^2 \times 750 \text{ cm} \]
Calculating this step-by-step:
1. Calculate \( (10.5)^2 = 110.25\)
2. Now substitute:
\[ V = \frac{22}{7} \times 110.25 \times 750 \]
3. Calculate \(110.25 \times 750 = 82687.5\)
4. Now multiply:
\[ V = \frac{22}{7} \times 82687.5 \]
5. Dividing \(82687.5\) by \(7\) gives approximately \(11812.5\).
6. Finally, calculate \(22 \times 11812.5 = 259875 \text{ cm}^3\).

### Final Answers:
(i) Curved Surface Area of the cylinder = \(49500 \text{ cm}^2\)
(ii) Volume of the cylinder = \(259875 \text{ cm}^3\)

---