Rectangle Area Calculation from Circle Wire

The problem asks for the area of a rectangle formed by reshaping a circular wire. The key is that the total length of the wire remains constant, meaning the circle's circumference equals the rectangle's perimeter.

Step 1: Calculate the Circle's Circumference

The diameter of the circle is given as $d = 49$ cm. The circumference ($C$) is calculated using the formula $C = \pi d$. Using the given value $\pi = \frac{22}{7}$:

$C = \frac{22}{7} \times 49$

$C = 22 \times 7$

$C = 154$ cm$

Step 2: Determine the Rectangle's Perimeter

Since the wire is reformed into a rectangle, the perimeter ($P$) of the rectangle is equal to the circumference of the circle:

$P = C = 154$ cm$

Step 3: Find the Rectangle's Dimensions

The sides of the rectangle are in the ratio 7 : 4. Let the sides be $7x$ and $4x$. The perimeter of a rectangle is given by $P = 2(\text{length} + \text{width})$.

$P = 2(7x + 4x)$

$154 = 2(11x)$

$154 = 22x$

Solving for $x$:

$x = \frac{154}{22}$

$x = 7$

Now, find the actual length and width:

• Length $l = 7x = 7 \times 7 = 49$ cm
• Width $w = 4x = 4 \times 7 = 28$ cm

Step 4: Calculate the Rectangle's Area

The area ($A$) of a rectangle is calculated by multiplying its length and width ($A = l \times w$).

$A = 49 \times 28$

$A = 1372$ sq cm$

Conclusion

The area of the rectangle is 1372 sq cm.