From a rectangular cardboard sheet 145 cm long and 32 cm broad ,42 circular plates each of diameter 8 cm have been cut out . Find the area of the remaining portion of the sheet.

Answer

To solve the problem step by step, we will calculate the area of the rectangular cardboard sheet, the area of one circular plate, the total area of the circular plates cut out, and finally, the area of the remaining portion of the sheet.

### Step 1: Calculate the area of the rectangular cardboard sheet.
The area of a rectangle is given by the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]

Given:
- Length = 145 cm
- Breadth = 32 cm

Calculating the area:
\[ \text{Area} = 145 \, \text{cm} \times 32 \, \text{cm} = 4640 \, \text{cm}^2 \]

### Step 2: Calculate the area of one circular plate.
The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]

Given:
- Diameter of the circular plate = 8 cm
- Radius \( r = \frac{\text{Diameter}}{2} = \frac{8}{2} = 4 \, \text{cm} \)

Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area} = \frac{22}{7} \times (4 \, \text{cm})^2 = \frac{22}{7} \times 16 \, \text{cm}^2 \]
\[ \text{Area} = \frac{352}{7} \, \text{cm}^2 \approx 50.29 \, \text{cm}^2 \]

### Step 3: Calculate the total area of 42 circular plates.
To find the total area of 42 circular plates:
\[ \text{Total Area} = \text{Area of one plate} \times \text{Number of plates} \]
\[ \text{Total Area} = \frac{352}{7} \, \text{cm}^2 \times 42 \]

Calculating:
\[ \text{Total Area} = \frac{352 \times 42}{7} \]
\[ = 352 \times 6 = 2112 \, \text{cm}^2 \]

### Step 4: Calculate the area of the remaining portion of the sheet.
To find the area of the remaining portion of the cardboard sheet:
\[ \text{Remaining Area} = \text{Area of rectangular sheet} - \text{Total Area of circular plates} \]
\[ \text{Remaining Area} = 4640 \, \text{cm}^2 - 2112 \, \text{cm}^2 \]
\[ \text{Remaining Area} = 2528 \, \text{cm}^2 \]

### Final Answer:
The area of the remaining portion of the sheet is **2528 cm²**.

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