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.
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