50 students of class IX planned a visit to an old age home and to spend the whole day with its inmates. Each one prepared a cylinderical flower vase using card board to gift the inmates. The radius of cylinder is 4.2 cm and the height is 11.2 cm.
What is the amount spent for purchasing the card board at the rate of 20 per `100m^(2)`.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the Curved Surface Area and Base Area of One Cylinder
The formula for the Curved Surface Area (CSA) of a cylinder is:
\[ \text{CSA} = 2\pi rh \]
The formula for the area of the base (circle) is:
\[ \text{Base Area} = \pi r^2 \]

Given:
- Radius \( r = 4.2 \, \text{cm} \)
- Height \( h = 11.2 \, \text{cm} \)

### Step 2: Calculate the Curved Surface Area
Substituting the values into the CSA formula:
\[ \text{CSA} = 2 \times \frac{22}{7} \times 4.2 \times 11.2 \]
Calculating the values step by step:
\[ = 2 \times \frac{22}{7} \times 4.2 \times 11.2 \]
\[ = 2 \times \frac{22 \times 4.2 \times 11.2}{7} \]
Calculating \( 22 \times 4.2 = 92.4 \)
\[ = 2 \times \frac{92.4 \times 11.2}{7} \]
Calculating \( 92.4 \times 11.2 = 1031.68 \)
\[ = 2 \times \frac{1031.68}{7} \]
Calculating \( \frac{1031.68}{7} \approx 147.38 \)
\[ = 2 \times 147.38 \approx 294.76 \, \text{cm}^2 \]

### Step 3: Calculate the Base Area
Now, calculate the base area:
\[ \text{Base Area} = \pi r^2 = \frac{22}{7} \times (4.2)^2 \]
Calculating \( (4.2)^2 = 17.64 \):
\[ = \frac{22}{7} \times 17.64 \]
Calculating \( \frac{22 \times 17.64}{7} \approx 55.44 \, \text{cm}^2
\]

### Step 4: Total Surface Area of One Cylinder
Now, add the Curved Surface Area and Base Area:
\[ \text{Total Area} = \text{CSA} + \text{Base Area} \]
\[ = 294.76 + 55.44 = 350.20 \, \text{cm}^2 \]

### Step 5: Calculate Total Area for 50 Cylinders
Now, multiply the area of one cylinder by the number of students (50):
\[ \text{Total Area for 50 Cylinders} = 50 \times 350.20 = 17510 \, \text{cm}^2 \]

### Step 6: Convert Area from cm² to m²
Since the cost is given per 100 m², we need to convert cm² to m²:
\[ 1 \, \text{m}^2 = 10,000 \, \text{cm}^2 \]
Thus,
\[ \text{Total Area in m}^2 = \frac{17510}{10000} = 1.751 \, \text{m}^2 \]

### Step 7: Calculate the Cost of Cardboard
The cost of cardboard is Rs. 20 for 100 m². Therefore, the cost per m² is:
\[ \text{Cost per m}^2 = \frac{20}{100} = 0.20 \, \text{Rs.} \]
Now, calculate the total cost:
\[ \text{Total Cost} = 1.751 \times 0.20 = 0.3502 \, \text{Rs.} \]

### Final Answer
The total amount spent for purchasing the cardboard is approximately Rs. 0.35.