1. **Problem statement:** The university owns a triangular piece of land with two sides measuring 650 m and 720 m, and the angle between them is 75 degrees. We need to find: (a) The area of the land. (b)(i) The total cost of constructing a circular fence touching all vertices (circumcircle). (b)(ii) Whether the university took over land belonging to others. 2. **Formula for area of a triangle given two sides and included angle:** $$\text{Area} = \frac{1}{2}ab\sin(C)$$ where $a=650$, $b=720$, and $C=75^\circ$. 3. **Calculate the area:** $$\text{Area} = \frac{1}{2} \times 650 \times 720 \times \sin(75^\circ)$$ Using $\sin(75^\circ) \approx 0.9659$, $$\text{Area} = 0.5 \times 650 \times 720 \times 0.9659 = 225,468 \text{ square meters (approx)}$$ 4. **Find the length of the third side using the Law of Cosines:** $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ $$c^2 = 650^2 + 720^2 - 2 \times 650 \times 720 \times \cos(75^\circ)$$ Using $\cos(75^\circ) \approx 0.2588$, $$c^2 = 422,500 + 518,400 - 2 \times 650 \times 720 \times 0.2588$$ $$c^2 = 940,900 - 242,227 = 698,673$$ $$c = \sqrt{698,673} \approx 836.0 \text{ meters}$$ 5. **Calculate the circumradius $R$ of the triangle:** Formula: $$R = \frac{abc}{4 \times \text{Area}}$$ $$R = \frac{650 \times 720 \times 836}{4 \times 225,468}$$ $$R = \frac{390,528,000}{901,872} \approx 433.1 \text{ meters}$$ 6. **Calculate the circumference of the circular fence:** $$\text{Circumference} = 2\pi R = 2 \times 3.1416 \times 433.1 \approx 2,720.5 \text{ meters}$$ 7. **Calculate the labor cost:** Labor cost per meter = 10 $$\text{Labor cost} = 2,720.5 \times 10 = 27,205$$ 8. **Convert construction materials cost from UGX to dollars:** Given materials cost = 10,567,800 UGX Exchange rate: 1 dollar = 3750 UGX $$\text{Materials cost in dollars} = \frac{10,567,800}{3750} = 2,818.08$$ 9. **Total cost of constructing the fence:** $$\text{Total cost} = \text{Labor cost} + \text{Materials cost} = 27,205 + 2,818.08 = 30,023.08$$ 10. **Did the university take over land belonging to others?** The circular fence passes through all vertices of the triangle, so it encloses exactly the triangular land owned by the university. Since the fence is circular and touches all vertices, it does not extend beyond the triangle, so no land outside the triangle is enclosed. Therefore, the university did not take over land belonging to others. **Final answers:** (a) Area $\approx 225,468$ square meters (b)(i) Total cost $\approx 30,023.08$ (b)(ii) No, the university did not take over land belonging to others.