1. **Problem statement:** Calculate the missing size of the circular ring given inner radius $r_i$, outer radius $r_a$, and area $A$ of the ring. 2. **Formula:** The area of a circular ring (annulus) is given by $$A = \pi (r_a^2 - r_i^2)$$ where $r_a$ is the outer radius and $r_i$ is the inner radius. 3. **Important rules:** - Radii must be in the same units before calculation. - Area units correspond to the square of the length units. 4. **Calculations:** **a)** Given $r_i = 7$ dm and $A = 34$ cm², find $r_a$. Convert $r_i$ to cm: $7$ dm $= 70$ cm. Use formula: $$34 = \pi (r_a^2 - 70^2)$$ $$r_a^2 = \frac{34}{\pi} + 4900$$ Calculate: $$r_a^2 = \frac{34}{3.1416} + 4900 \approx 10.82 + 4900 = 4910.82$$ $$r_a = \sqrt{4910.82} \approx 70.07 \text{ cm} = 7.007 \text{ dm}$$ **b)** Given $r_a = ?$, $r_i = 8.6$ cm, and $A = 168.78$ cm², find $r_a$. Use formula: $$168.78 = \pi (r_a^2 - 8.6^2)$$ $$r_a^2 = \frac{168.78}{\pi} + 8.6^2 = \frac{168.78}{3.1416} + 73.96 \approx 53.74 + 73.96 = 127.7$$ $$r_a = \sqrt{127.7} \approx 11.3 \text{ cm}$$ **c)** Given $r_a = 49.5$ m, $r_i = ?$, and $A = 5321.86$ m², find $r_i$. Use formula: $$5321.86 = \pi (49.5^2 - r_i^2)$$ $$r_i^2 = 49.5^2 - \frac{5321.86}{\pi} = 2450.25 - \frac{5321.86}{3.1416} \approx 2450.25 - 1694.9 = 755.35$$ $$r_i = \sqrt{755.35} \approx 27.49 \text{ m}$$ 5. **Summary of answers:** - a) $r_a \approx 7.007$ dm - b) $r_a \approx 11.3$ cm - c) $r_i \approx 27.49$ m These calculations show how to find the missing radius using the area formula for circular rings.