From a circular piece of cardboard of radius 3 cm, two sectors of 40° each have been cut off. The area of the remaining portion is : $(Use \pi = 22/7)$

Question

Accepted Answer

To determine the area of the remaining portion of the circular piece of cardboard after two sectors of 40° each have been cut off, we need to follow these steps:

1. **Calculate the Total Area of the Circle:**
   The formula for the area of a circle is $ A = \pi r^2 $.
   Given the radius $ r = 3 $ cm, and using $ \pi = \frac{22}{7} $, the total area of the circle is:
   $$
   A = \frac{22}{7} \times 3^2 = \frac{22}{7} \times 9 = \frac{198}{7} \text{ cm}^2
   $$

2. **Calculate the Area of One Sector:**
   The area of a sector of a circle is given by the formula $ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 $, where $ \theta $ is the angle of the sector.
   For one sector with $ \theta = 40^\circ $:
   $$
   \text{Area of one sector} = \frac{40}{360} \times \frac{22}{7} \times 9 = \frac{1}{9} \times \frac{198}{7} = \frac{22}{7} \text{ cm}^2
   $$

3. **Calculate the Total Area of the Two Sectors:**
   Since there are two sectors of 40° each, the total area of the two sectors is:
   $$
   2 \times \frac{22}{7} = \frac{44}{7} \text{ cm}^2
   $$

4. **Calculate the Area of the Remaining Portion:**
   Subtract the total area of the two sectors from the total area of the circle:
   $$
   \text{Remaining area} = \frac{198}{7} - \frac{44}{7} = \frac{154}{7} = 22 \text{ cm}^2
   $$

Thus, the area of the remaining portion of the cardboard is $ 22 \text{ cm}^2 $, which corresponds to Option 2.

**Explanation for Incorrect Options:**

- **Option 1: $11 \text{ cm}^2$**
  - This option is incorrect because the calculation shows that the remaining area is $ 22 \text{ cm}^2 $, not $ 11 \text{ cm}^2 $.

- **Option 3: $33 \text{ cm}^2$**
  - This option is incorrect because the calculated remaining area is $ 22 \text{ cm}^2 $, which is less than $ 33 \text{ cm}^2 $.

- **Option 4: $44 \text{ cm}^2$**
  - This option is incorrect because the remaining area is $ 22 \text{ cm}^2 $, which is exactly half of $ 44 \text{ cm}^2 $.

Therefore, based on the calculations and logical reasoning, Option 2 is indeed the correct answer.