1. **Problem Statement:** Two vertical poles each 18 m high stand at points A and B along a straight horizontal road. A footpath meets the road at B from a point C on the footpath 43 m from the road. The angles of elevation from C to the tops of poles at A and B are 14.6° and 16.4° respectively. We need to find: (a) The distances AC and BC (b) The distance between the poles A and B 2. **Understanding the setup:** - The poles are vertical, so their heights are perpendicular to the road. - The footpath meets the road at B, and C is 43 m from the road along the footpath. - Angles of elevation are measured from point C to the tops of the poles. 3. **Key formulas and rules:** - Use the tangent of the angle of elevation: $$\tan(\theta) = \frac{\text{height}}{\text{horizontal distance}}$$ - Horizontal distances from C to the bases of poles A and B can be found using the tangent formula. - The distance BC is given as 43 m. 4. **Calculate horizontal distances from C to poles:** - Let $x$ be the horizontal distance from C to B along the road (which is 0 since B is on the road), and $d$ be the distance from B to A along the road. - From C to B, the horizontal distance is 0 (since B is on the road), but the footpath length BC is 43 m (distance from C to road). 5. **Calculate horizontal distances from C to poles' bases:** - For pole B (height 18 m, angle 16.4°): $$\tan(16.4^\circ) = \frac{18}{\text{distance from C to B}}$$ Since C is 43 m from the road, the horizontal distance from C to B along the road is 0, but the footpath length is 43 m. - For pole A (height 18 m, angle 14.6°): $$\tan(14.6^\circ) = \frac{18}{\text{distance from C to A}}$$ 6. **Calculate distances:** - Distance from C to B (horizontal) = $$\frac{18}{\tan(16.4^\circ)} = \frac{18}{0.294} = 61.2\text{ m}$$ - Distance from C to A (horizontal) = $$\frac{18}{\tan(14.6^\circ)} = \frac{18}{0.261} = 69.0\text{ m}$$ 7. **Calculate distance AB (distance between poles):** - Since A and B lie on the road, and distances from C to A and C to B are known, the distance AB is: $$AB = AC - BC = 69.0 - 61.2 = 7.8\text{ m}$$ **Final answers:** (a) Distance AC = 69.0 m, Distance BC = 61.2 m (b) Distance between poles AB = 7.8 m All distances are rounded to one decimal place as requested.