Question

The diameter of a circular garden is \(140\) m. Its area is equal to a rectangular field whose sides are in the ratio \(11:7\). The perimeter (in m) of the rectangular field is (take \(\pi = \frac{22}{7}\)):

• A. \(360\sqrt{2}\)
• B. \(180\sqrt{2}\)
• C. \(120\sqrt{2}\)
• D. \(270\sqrt{2}\)

Hint:

Remember:

• Area of circle: \[ \pi r^2 \]
• If rectangle sides are in ratio \(a:b\): \[ \text{Sides} = ax,\; bx \]
• Perimeter: \[ 2(l+b) \]

Answer 1

The Correct Option is D

Concept: Area of a circle: \[ A = \pi r^2 \] For a rectangle whose sides are in ratio \(11:7\), let the sides be: \[ 11x \quad \text{and} \quad 7x \] Then: \[ \text{Area} = 11x \times 7x \]

Step 1: Find the area of the circular garden. Given: \[ \text{Diameter} = 140\,\text{m} \] Hence radius: \[ r = \frac{140}{2} = 70\,\text{m} \] Using: \[ A = \pi r^2 \] \[ A = \frac{22}{7}\times 70 \times 70 \] \[ A = 22 \times 10 \times 70 \] \[ A = 15400\,\text{m}^2 \]

Step 2: Form the area equation for the rectangle. Sides of rectangle are in ratio: \[ 11:7 \] Let sides be: \[ 11x \quad \text{and} \quad 7x \] Area: \[ 11x \times 7x = 77x^2 \] Since areas are equal: \[ 77x^2 = 15400 \] \[ x^2 = \frac{15400}{77} \] \[ x^2 = 200 \] \[ x = 10\sqrt{2} \]

Step 3: Calculate the perimeter of rectangle. Perimeter: \[ P = 2(l+b) \] \[ P = 2(11x + 7x) \] \[ P = 2(18x) \] \[ P = 36x \] Substituting: \[ x = 10\sqrt{2} \] \[ P = 36 \times 10\sqrt{2} \] \[ P = 360\sqrt{2} \] Therefore, \[ \boxed{360\sqrt{2}} \]