A circular pond has area equal to 616 m². A circular stage is made at the centre of the pond whose radius is equal to half the radius of the pond. What is the area where water is present?

### Concept & Proportionality When comparing two circles, if the radius of the smaller circle is a fraction of the larger circle, its area scales by the square of that fraction. $$\frac{A_{\text{small}}}{A_{\text{large}}} = \left(\frac{r_{\text{small}}}{r_{\text{large}}}\right)^2$$ ### Step-by-Step Solution * Given: Total Area of the pond = 616 m². * Given: Radius of the stage ($r_{\text{stage}}$) = $\frac{1}{2}$ Radius of the pond ($r_{\text{pond}}$). * Since the radius is halved, the area of the stage is $(\frac{1}{2})^2 = \frac{1}{4}$ of the pond's total area. * Calculate the area of the stage: $\text{Area}_{\text{stage}} = \frac{1}{4} \times 616 = 154$ m². * Calculate the area where water is present by subtracting the stage area from the total area: $\text{Area}_{\text{water}} = \text{Total Area} - \text{Area}_{\text{stage}}$ $\text{Area}_{\text{water}} = 616 - 154 = 462$ m². ### Exam Strategy & Shortcut If the stage takes up $\frac{1}{4}$ of the pond's area, then the water must take up the remaining $\frac{3}{4}$ of the area. Simply multiply the total area by $\frac{3}{4}$: $\frac{3}{4} \times 616 = 3 \times 154 = 462$ sq. m. This entirely avoids calculating the actual radii in meters. ### Common Pitfall Setting the equation $\pi r^2 = 616$, solving for $r = 14$, finding the new radius $r=7$, squaring it, and multiplying by $\pi$ again. While this works, it wastes valuable time compared to using direct area scaling ratios. ### Final Answer Therefore, the correct answer is **462 sq. m**.