A circular wire of radius 42 cm is bent in the form of a rectangle whose sides are in the ratio of `6 :5`. The smaller side of the reactangle is (Take `pi = (22)/(7)`)

Options

A. 60 cm
B. 30 cm
C. 25 cm
D. 36 cm

Answer

To solve the problem, we need to find the smaller side of a rectangle formed by bending a circular wire of radius 42 cm. The sides of the rectangle are in the ratio of 6:5. Let's go through the solution step by step.

### Step 1: Calculate the circumference of the circular wire
The circumference \( C \) of a circle is given by the formula:
\[ C = 2 \pi r \]
where \( r \) is the radius of the circle. Given that the radius \( r = 42 \) cm and taking \( \pi = \frac{22}{7} \), we can substitute these values into the formula:
\[ C = 2 \times \frac{22}{7} \times 42 \]

### Step 2: Simplify the circumference calculation
Now, let's calculate the circumference:
\[ C = 2 \times \frac{22}{7} \times 42 = \frac{2 \times 22 \times 42}{7} \]
Calculating \( 2 \times 22 = 44 \):
\[ C = \frac{44 \times 42}{7} \]
Now, calculate \( 44 \times 42 = 1848 \):
\[ C = \frac{1848}{7} = 264 \text{ cm} \]

### Step 3: Set up the perimeter of the rectangle
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + b) \]
where \( l \) is the length and \( b \) is the breadth. According to the problem, the sides of the rectangle are in the ratio of 6:5. We can express the length and breadth in terms of a variable \( x \):
\[ l = 6x \quad \text{and} \quad b = 5x \]
Thus, the perimeter can be expressed as:
\[ P = 2(6x + 5x) = 2(11x) = 22x \]

### Step 4: Equate the perimeter of the rectangle to the circumference of the wire
Since the wire is bent to form the rectangle, the perimeter of the rectangle is equal to the circumference of the wire:
\[ 22x = 264 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{264}{22} = 12 \]

### Step 6: Find the smaller side of the rectangle
Now that we have \( x \), we can find the smaller side (breadth) of the rectangle:
\[ b = 5x = 5 \times 12 = 60 \text{ cm} \]

### Final Answer
The smaller side of the rectangle is **60 cm**.

---