Semi-circular lawns are attached to the edges of a rectangular field measuring 42 m × 35 m. The area of the total field is

### Concept & Geometry The total area of the field consists of the area of the central rectangle plus the areas of four semi-circles attached to its sides. Area of a rectangle = $Length \times Width$ Area of a semi-circle = $\frac{1}{2} \pi r^2$ ### Step-by-Step Solution **1. Area of the rectangular field:** * Length $l = 42 \text{ m}$, Width $w = 35 \text{ m}$. * Area $= 42 \times 35 = 1470 \text{ m}^2$. **2. Area of semi-circles on the length sides (42 m):** * There are two such semi-circles. Together they form one full circle of diameter 42 m. * Radius $r_1 = \frac{42}{2} = 21 \text{ m}$. * Area $= \pi r_1^2 = \frac{22}{7} \times 21 \times 21 = 22 \times 3 \times 21 = 1386 \text{ m}^2$. **3. Area of semi-circles on the width sides (35 m):** * There are two such semi-circles. Together they form one full circle of diameter 35 m. * Radius $r_2 = \frac{35}{2} = 17.5 \text{ m}$. * Area $= \pi r_2^2 = \frac{22}{7} \times 17.5 \times 17.5 = \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2} = 11 \times 5 \times 17.5 = 962.5 \text{ m}^2$. **4. Total Area:** * Total Area $= 1470 + 1386 + 962.5 = 3818.5 \text{ m}^2$. ### Exam Strategy & Shortcut Realize that the four semi-circles combine into two full circles. So the total area is simply: $lw + \pi(\frac{l}{2})^2 + \pi(\frac{w}{2})^2$. To speed up calculations, notice that $1470 + 1386 = 2856$. Adding a number slightly less than $1000$ to $2856$ yields around $3800$. Option (b) is the only reasonable choice. ### Common Pitfall A common mistake is treating the attached regions as full circles rather than semi-circles when calculating individual side attachments, doubling the extra area erroneously. ### Final Answer Therefore, the correct answer is **3818.5 m²**.