A circular grassy plot of land, 42 m in diameter, has a path 3.5 m wide running around it outside. The cost of gravelling the path at ₹ 4 per square metre is

### Concept & Formula Calculate the area of the circular path by finding the difference in areas between the outer circle and inner circle, then multiply by the cost rate. $$\text{Area of path} = \pi(R^2 - r^2) = \pi(R - r)(R + r)$$ $$\text{Total Cost} = \text{Area} \times \text{Rate}$$ ### Step-by-Step Solution * **Given:** * Diameter of plot = $42\text{ m}$ * Inner radius ($r$) = $\frac{42}{2} = 21\text{ m}$ * Width of path = $3.5\text{ m}$ * Outer radius ($R$) = $21 + 3.5 = 24.5\text{ m}$ * **Find the area of the path:** $$\text{Area} = \frac{22}{7} \times (24.5^2 - 21^2)$$ $$\text{Area} = \frac{22}{7} \times (24.5 - 21) \times (24.5 + 21)$$ $$\text{Area} = \frac{22}{7} \times 3.5 \times 45.5$$ $$\text{Area} = 22 \times 0.5 \times 45.5$$ $$\text{Area} = 11 \times 45.5 = 500.5\text{ sq m}$$ * **Calculate total cost:** $$\text{Total Cost} = 500.5 \times 4 = \text{₹ } 2002$$ ### Exam Strategy & Shortcut Delay full multiplication until the end. Notice that $3.5 = \frac{7}{2}$. $\text{Area} = \frac{22}{7} \times \frac{7}{2} \times 45.5 = 11 \times 45.5$. Then directly multiply by $4$: $11 \times 45.5 \times 4 = 44 \times 45.5 = 2002$. ### Common Pitfall Adding the path width ($3.5\text{ m}$) to the diameter ($42\text{ m}$) instead of the radius ($21\text{ m}$), which skews the outer boundary measurement entirely. ### Final Answer Therefore, the correct answer is **₹ 2002**.