The circle of largest area is cut from a rectangular piece of card-board with dimensions 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 of finding the ratio between the area of the circle cut from a rectangular piece of cardboard and the area of the remaining cardboard, we will follow these steps:

### Step 1: Calculate the area of the rectangular cardboard.
The dimensions of the rectangular piece of cardboard are given as 55 cm and 42 cm.

**Formula for the area of a rectangle:**
\[ \text{Area} = \text{Length} \times \text{Breadth} \]

**Calculation:**
\[ \text{Area of rectangle} = 55 \, \text{cm} \times 42 \, \text{cm} = 2310 \, \text{cm}^2 \]

### Step 2: Determine the diameter and radius of the largest circle that can be cut from the rectangle.
The largest circle that can be cut from the rectangle will have a diameter equal to the smaller side of the rectangle, which is 42 cm.

**Calculation of radius:**
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{42 \, \text{cm}}{2} = 21 \, \text{cm} \]

### Step 3: Calculate the area of the circle.
**Formula for the area of a circle:**
\[ \text{Area} = \pi r^2 \]

Using \(\pi \approx \frac{22}{7}\):
\[ \text{Area of circle} = \frac{22}{7} \times (21 \, \text{cm})^2 \]

**Calculation:**
\[ = \frac{22}{7} \times 441 \, \text{cm}^2 \]
\[ = \frac{22 \times 441}{7} = \frac{9702}{7} = 1386 \, \text{cm}^2 \]

### Step 4: Calculate the area of the remaining cardboard.
**Formula for remaining area:**
\[ \text{Remaining Area} = \text{Total Area} - \text{Area of Circle} \]

**Calculation:**
\[ \text{Remaining Area} = 2310 \, \text{cm}^2 - 1386 \, \text{cm}^2 = 924 \, \text{cm}^2 \]

### Step 5: Calculate the ratio of the area of the circle to the area of the remaining cardboard.
**Formula for ratio:**
\[ \text{Ratio} = \frac{\text{Area of Circle}}{\text{Remaining Area}} \]

**Calculation:**
\[ \text{Ratio} = \frac{1386 \, \text{cm}^2}{924 \, \text{cm}^2} \]

To simplify the ratio:
\[ = \frac{1386 \div 462}{924 \div 462} = \frac{3}{2} \]

### Final Answer:
The ratio between the area of the circle cut and the area of the remaining cardboard is \( \frac{3}{2} \).

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