Let's solve the given problem step-by-step.

Problem:

From a circular card sheet of radius 14 cm, two circles of radius 3.5 cm each and a rectangle of length 3 cm and breadth 1 cm are removed. Find the area of the remaining sheet. Use $\pi = \frac{22}{7}$.

Step 1: Find the area of the original circular card sheet.

The formula for the area of a circle is:

$\text{Area} = \pi r^2$

Here, the radius $r = 14$ cm. Substituting the values:

$\text{Area of the circular card} = \frac{22}{7} \times 14^2$

First, calculate $14^2 = 196$:

$\text{Area} = \frac{22}{7} \times 196 = 22 \times 28 = 616 \, \text{cm}^2$

So, the total area of the circular card is 616 cm².

Step 2: Find the total area removed.

(a) Area of two small circles

The radius of each smaller circle is $r = 3.5$ cm. The area of one circle is:

$\text{Area of one circle} = \pi r^2 = \frac{22}{7} \times (3.5)^2$

First, calculate $(3.5)^2 = 12.25$:

$\text{Area of one circle} = \frac{22}{7} \times 12.25 = 22 \times 1.75 = 38.5 \, \text{cm}^2$

Thus, the total area of two circles is:

$\text{Total area of two circles} = 2 \times 38.5 = 77 \, \text{cm}^2$

(b) Area of the rectangle

The length of the rectangle is $3$ cm, and the breadth is $1$ cm. The area is:

$\text{Area of rectangle} = \text{length} \times \text{breadth} = 3 \times 1 = 3 \, \text{cm}^2$

Total area removed

The total area removed is:

$\text{Total area removed} = 77 + 3 = 80 \, \text{cm}^2$

Step 3: Find the area of the remaining sheet.

The area of the remaining sheet is:

$\text{Remaining area} = \text{Original area} - \text{Removed area}$

Substituting the values:

$\text{Remaining area} = 616 - 80 = 536 \, \text{cm}^2$

Final Answer:

The area of the remaining sheet is:

$\boxed{536 \, \text{cm}^2}$

Would you like a further breakdown of any part of the solution? Let me know!

Here are 5 related questions to explore further:

1. How would the remaining area change if one more circle of radius 3.5 cm were removed?
2. If the rectangle had dimensions 5 cm by 2 cm, how would the solution change?
3. What is the perimeter of the original circular card sheet?
4. How can this problem be solved if the value of $\pi$ is approximated to 3.14 instead of $\frac{22}{7}$?
5. If only one circle and the rectangle were removed, what would be the remaining area?

Tip:

For problems involving areas of composite shapes, always break the problem into smaller parts, calculate each part separately, and then combine the results systematically.