A circle of largest area is cut from a rectangular piece of card-board with dimension 55 cm and 42 cm. Find the ratio between the area of the circle cut and the area of the remaining card-board.

Answer

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

### Step 1: Determine the dimensions of the rectangle
The dimensions of the rectangular piece of cardboard are given as:
- Length = 55 cm
- Breadth = 42 cm

### Step 2: Find the diameter of the largest circle that can be cut from the rectangle
The largest circle that can be cut from the rectangle will have its diameter equal to the smaller side of the rectangle. Therefore:
- Diameter of the circle = 42 cm

### Step 3: Calculate the radius of the circle
The radius (r) of the circle is half of the diameter:
\[ r = \frac{\text{Diameter}}{2} = \frac{42 \text{ cm}}{2} = 21 \text{ cm} \]

### Step 4: Calculate the area of the circle
The area (A) of the circle can be calculated using the formula:
\[ A = \pi r^2 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times (21 \text{ cm})^2 \]
\[ A = \frac{22}{7} \times 441 \text{ cm}^2 \]
\[ A = \frac{9702}{7} \text{ cm}^2 \approx 1386 \text{ cm}^2 \]

### Step 5: Calculate the area of the rectangle
The area (A_r) of the rectangle is calculated using the formula:
\[ A_r = \text{Length} \times \text{Breadth} \]
\[ A_r = 55 \text{ cm} \times 42 \text{ cm} \]
\[ A_r = 2310 \text{ cm}^2 \]

### Step 6: Calculate the area of the remaining cardboard
The area of the remaining cardboard after cutting out the circle is:
\[ \text{Area of remaining cardboard} = \text{Area of rectangle} - \text{Area of circle} \]
\[ \text{Area of remaining cardboard} = 2310 \text{ cm}^2 - 1386 \text{ cm}^2 \]
\[ \text{Area of remaining cardboard} = 924 \text{ cm}^2 \]

### Step 7: Find the ratio of the area of the circle to the area of the remaining cardboard
The ratio (R) is given by:
\[ R = \frac{\text{Area of circle}}{\text{Area of remaining cardboard}} \]
\[ R = \frac{1386 \text{ cm}^2}{924 \text{ cm}^2} \]
To simplify this ratio:
\[ R = \frac{1386 \div 462}{924 \div 462} = \frac{3}{2} \]

### Final Answer
The ratio between the area of the circle cut and the area of the remaining cardboard is:
\[ \text{Ratio} = 3 : 2 \]

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