A circular copper wire of radius $7\text{cm}$ is bent to form a rectangle. If the breadth and length of the rectangle are in the ratio of $4:7$ respectively, what is the breadth of the rectangle? (in $\text{cm}$)

When a wire is bent from a circular shape to a rectangular shape, its total length remains constant. This means the circumference of the circle is equal to the perimeter of the rectangle.
Circumference of the circle:

$$C=2\pi r=2\times \frac{22}{7}\times 7=44\text{cm}$$

Let the breadth and length of the rectangle be $4x$ and $7x$ respectively.
Perimeter of the rectangle:

$$P=2(\text{length}+\text{breadth})=2(7x+4x)=2(11x)=22x$$

Equating the perimeter to the circumference:

$$22x=44⟹x=2$$

Therefore, the breadth of the rectangle is:

$$\text{Breadth}=4x=4\times 2=8\text{cm}$$