1. **Problem statement:** Given quadrilateral ABCD with sides BC = 192 m, CD = 287.9 m, BD = 168 m, and AD = 205.8 m, we need to: (a)(i) Calculate angle CBD and show it rounds to 106.0°. (ii) Find the bearing of C from B given the bearing of D from B is 038°. (iii) Calculate the bearing of D from A given A is due east of B. (b)(i) Calculate the area of triangle BCD. (ii) Calculate the cost Tomas pays for the triangular part BCD at 35750 per hectare. --- 2. **(a)(i) Calculate angle CBD:** Use the cosine rule in triangle BCD to find angle CBD (angle at B between points C and D): $$\cos(\angle CBD) = \frac{BC^2 + BD^2 - CD^2}{2 \times BC \times BD}$$ Substitute values: $$\cos(\angle CBD) = \frac{192^2 + 168^2 - 287.9^2}{2 \times 192 \times 168}$$ Calculate numerator: $$192^2 = 36864, \quad 168^2 = 28224, \quad 287.9^2 \approx 82845.61$$ $$36864 + 28224 - 82845.61 = 65088 - 82845.61 = -17757.61$$ Calculate denominator: $$2 \times 192 \times 168 = 64512$$ Calculate cosine: $$\cos(\angle CBD) = \frac{-17757.61}{64512} \approx -0.2753$$ Find angle: $$\angle CBD = \cos^{-1}(-0.2753) \approx 106.0^\circ$$ This matches the required rounding. --- 3. **(a)(ii) Bearing of C from B:** Given bearing of D from B is 038° and angle CBD = 106.0°, the bearing of C from B is: $$\text{Bearing of C} = 038^\circ + 106.0^\circ = 144.0^\circ$$ --- 4. **(a)(iii) Bearing of D from A:** Since A is due east of B, the bearing of B from A is 270° (west). The bearing of D from B is 038°, so angle ABD is 38° north of east. Using the given distances and bearings, the bearing of D from A is: $$\text{Bearing of D from A} = 360^\circ - 38^\circ = 322^\circ$$ --- 5. **(b)(i) Area of triangle BCD:** Use Heron's formula: $$s = \frac{BC + CD + BD}{2} = \frac{192 + 287.9 + 168}{2} = \frac{647.9}{2} = 323.95$$ Area: $$\sqrt{s(s - BC)(s - CD)(s - BD)} = \sqrt{323.95(323.95 - 192)(323.95 - 287.9)(323.95 - 168)}$$ Calculate each term: $$323.95 - 192 = 131.95$$ $$323.95 - 287.9 = 36.05$$ $$323.95 - 168 = 155.95$$ Calculate product: $$323.95 \times 131.95 \times 36.05 \times 155.95 \approx 240,000,000$$ Area: $$\sqrt{240,000,000} \approx 15491.9 \text{ m}^2$$ --- 6. **(b)(ii) Cost calculation:** Convert area to hectares: $$\frac{15491.9}{10,000} = 1.54919 \text{ hectares}$$ Calculate cost: $$1.54919 \times 35750 = 55388.4$$ Rounded to nearest 100: $$55300$$ --- **Final answers:** (a)(i) $106.0^\circ$ (a)(ii) $144.0^\circ$ (a)(iii) $322^\circ$ (b)(i) $15491.9$ m$^2$ (b)(ii) $55300$