The area of a circular field is equal to the area of a rectangular field. The ratio of the length and the breadth of the rectangular field is 14 : 11 respectively and perimeter is 100 metres. What is the diameter of the circular field?

### Concept & Area Equivalence Use the given ratio and perimeter to find the exact dimensions of the rectangle. Then, equate its area to the circle's area formula to find the radius and diameter. $$ \text{Perimeter of Rectangle} = 2(l + b) $$ $$ \text{Area of Rectangle} = l \times b $$ $$ \text{Area of Circle} = \pi r^2 $$ ### Step-by-Step Solution * Given: Ratio of length to breadth $l : b = 14 : 11$. Let $l = 14x$ and $b = 11x$. * Given: Perimeter = $100\text{ m}$. * Set up equation: $2(14x + 11x) = 100 \Rightarrow 2(25x) = 100 \Rightarrow 50x = 100 \Rightarrow x = 2$. * Find dimensions: Length $l = 14 \times 2 = 28\text{ m}$, Breadth $b = 11 \times 2 = 22\text{ m}$. * Calculate Rectangle Area: $28 \times 22 = 616\text{ m}^2$. * Equate to Circle Area: $\pi r^2 = 616$. * Solve for radius $r$: $\frac{22}{7} \times r^2 = 616 \Rightarrow r^2 = \frac{616 \times 7}{22} = 28 \times 7 = 196$. * $r = \sqrt{196} = 14\text{ m}$. * Calculate Diameter: $d = 2r = 2 \times 14 = 28\text{ m}$. ### Exam Strategy & Shortcut Instead of multiplying $28 \times 22$ to get $616$, leave it as $28 \times 22$. When equating to $\pi r^2$, write $\frac{22}{7} \times r^2 = 28 \times 22$. The $22$ on both sides cancels out immediately, leaving $r^2 = 28 \times 7$, which is faster to evaluate. ### Common Pitfall Stopping at $r = 14\text{ m}$ and selecting option (a), forgetting that the question specifically asks for the *diameter* of the circular field, not the radius. ### Final Answer Therefore, the correct answer is **28 m**.