A circular wire of radius 4.2 m is cut and bent in the form of a rectangle whose longer side is 20% more than its shorter side . The longer side of the rectangle is :

Options

A. 7.2 m
B. 72 cm
C. 8 m
D. none of these

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### 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. Given that the radius \( r = 4.2 \) m, we can substitute this value into the formula:
\[ C = 2 \times \frac{22}{7} \times 4.2 \]

### Step 2: Simplify the circumference calculation
Calculating the above expression:
\[ C = 2 \times \frac{22}{7} \times 4.2 = \frac{44}{7} \times 4.2 \]
To simplify \( 4.2 \):
\[ 4.2 = \frac{42}{10} = \frac{21}{5} \]
Thus,
\[ C = \frac{44}{7} \times \frac{21}{5} = \frac{924}{35} = 26.4 \text{ m} \]

### Step 3: Set up the rectangle's dimensions
Let the shorter side of the rectangle be \( x \). According to the problem, the longer side is 20% more than the shorter side. Therefore, the longer side can be expressed as:
\[ \text{Longer side} = x + 0.2x = 1.2x \]

### Step 4: Write the perimeter of the rectangle
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(\text{Length} + \text{Breadth}) = 2(x + 1.2x) = 2(2.2x) = 4.4x \]

### Step 5: Set the circumference equal to the perimeter
Since the wire is bent to form the rectangle, the circumference of the circle is equal to the perimeter of the rectangle:
\[ 4.4x = 26.4 \]

### Step 6: Solve for \( x \)
To find \( x \):
\[ x = \frac{26.4}{4.4} = 6 \text{ m} \]

### Step 7: Calculate the longer side
Now that we have \( x \), we can find the longer side:
\[ \text{Longer side} = 1.2x = 1.2 \times 6 = 7.2 \text{ m} \]

### Final Answer
The longer side of the rectangle is \( 7.2 \) m.

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