Question: Barangay Information Assume that Marikina City is located at the origin $$(0,0)$$ of the Cartesian plane. Barangay Malanday: The center is $$1,239.5m$$ west and $$131.9m$$ north of Marikina City. All residents within $$872.5m$$ of the center must be warned of possible flooding. Barangay Tumana: The center is $$812.3m$$ west and $$1,167.3m$$ north of Marikina City. All residents within $$877.5m$$ of the center must be warned of possible flooding. Barangay Nangka: The center is $$447.5m$$ east and $$2,152m$$ north of Marikina City. All residents within $$1,086.7m$$ of the center must be warned of possible flooding. Task 5: Designing the Speaker Station Network Determine strategic locations for high-powered speaker stations. Propose locations for speaker stations that can effectively serve the barangays. Determine how many speaker stations are needed. Write the equation of the coverage area for each speaker station using a $$500$$-meter radius. Graph the speaker stations together with the barangay coverage areas

1. **State the problem:** We need to design a network of speaker stations to cover the barangays Malanday, Tumana, and Nangka, each with given centers and flood warning radii. Each speaker station covers a radius of $$500$$ meters. 2. **Identify barangay centers and flood radii:** - Malanday center: $$(-1239.5, 131.9)$$ with flood radius $$872.5$$ m - Tumana center: $$(-812.3, 1167.3)$$ with flood radius $$877.5$$ m - Nangka center: $$(447.5, 2152)$$ with flood radius $$1086.7$$ m 3. **Speaker station coverage:** Each speaker station covers a circle of radius $$500$$ m. The equation of coverage area for a speaker station at $$ (h,k) $$ is: $$ (x - h)^2 + (y - k)^2 = 500^2 = 250000 $$ 4. **Determine strategic locations:** - To cover each barangay effectively, place speaker stations near the barangay centers. - Since flood warning radii are larger than speaker coverage, multiple stations may be needed per barangay. 5. **Calculate number of speaker stations per barangay:** - For Malanday, radius $$872.5$$ m, number of stations needed $$= \lceil \frac{872.5}{500} \rceil = 2$$ - For Tumana, radius $$877.5$$ m, number of stations needed $$= \lceil \frac{877.5}{500} \rceil = 2$$ - For Nangka, radius $$1086.7$$ m, number of stations needed $$= \lceil \frac{1086.7}{500} \rceil = 3$$ 6. **Proposed speaker station locations:** - Malanday: Place 2 stations along the line from center outward, e.g., at $$(-1239.5, 131.9)$$ and $$(-739.5, 131.9)$$ - Tumana: Place 2 stations, e.g., at $$(-812.3, 1167.3)$$ and $$(-312.3, 1167.3)$$ - Nangka: Place 3 stations, e.g., at $$(447.5, 2152)$$, $$(947.5, 2152)$$, and $$(1447.5, 2152)$$ 7. **Equations of coverage areas for each speaker station:** - Malanday stations: - $$ (x + 1239.5)^2 + (y - 131.9)^2 = 250000 $$ - $$ (x + 739.5)^2 + (y - 131.9)^2 = 250000 $$ - Tumana stations: - $$ (x + 812.3)^2 + (y - 1167.3)^2 = 250000 $$ - $$ (x + 312.3)^2 + (y - 1167.3)^2 = 250000 $$ - Nangka stations: - $$ (x - 447.5)^2 + (y - 2152)^2 = 250000 $$ - $$ (x - 947.5)^2 + (y - 2152)^2 = 250000 $$ - $$ (x - 1447.5)^2 + (y - 2152)^2 = 250000 $$ 8. **Summary:** - Total speaker stations needed: $$2 + 2 + 3 = 7$$ - Each station covers a $$500$$ m radius circle. 9. **Graphing:** - Plot barangay centers and their flood warning circles. - Plot speaker stations and their coverage circles. This design ensures all residents within flood warning radii are covered by at least one speaker station.