A circular wire of diameter 112 cm is cut and bent in the form of a rectangle whose sides are in the ratio of 9 : 7. The smaller side of the rectangle

Options

A. 77 cm
B. 97 cm
C. 67 cm
D. 87 cm

Answer

To solve the problem step-by-step, we will follow these instructions:

### Step 1: Find the circumference of the circular wire.
The diameter of the circular wire is given as 112 cm.

To find the circumference (C) of a circle, we use the formula:
\[ C = \pi \times d \]
Where \( d \) is the diameter.

Substituting the given diameter:
\[ C = \pi \times 112 \]
Using \( \pi \approx \frac{22}{7} \):
\[ C = \frac{22}{7} \times 112 \]

### Step 2: Calculate the circumference.
Calculating:
\[ C = \frac{22 \times 112}{7} = \frac{2464}{7} = 352 \text{ cm} \]

### Step 3: Set up the perimeter of the rectangle.
The wire is bent to form a rectangle whose sides are in the ratio of 9:7. Let the length (L) be \( 9x \) and the breadth (B) be \( 7x \).

The perimeter (P) of the rectangle is given by:
\[ P = 2(L + B) = 2(9x + 7x) = 2(16x) = 32x \]

### Step 4: Set the perimeter equal to the circumference.
Since the perimeter of the rectangle is equal to the circumference of the circle:
\[ 32x = 352 \]

### Step 5: Solve for \( x \).
Dividing both sides by 32:
\[ x = \frac{352}{32} = 11 \text{ cm} \]

### Step 6: Find the dimensions of the rectangle.
Now that we have \( x \), we can find the dimensions of the rectangle:
- Length \( L = 9x = 9 \times 11 = 99 \text{ cm} \)
- Breadth \( B = 7x = 7 \times 11 = 77 \text{ cm} \)

### Step 7: Identify the smaller side of the rectangle.
The smaller side of the rectangle is:
\[ B = 77 \text{ cm} \]

### Final Answer:
The smaller side of the rectangle is **77 cm**.

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