1. **Problem statement:** A vertical extension is constructed on top of a building on level ground. From point P on the ground, the angle of elevation to the top of the extension is 32° and to the bottom of the extension is 29°. Point Q is 53 m closer to the building than P, and the angle of elevation to the top of the extension from Q is 36°. We need to find the height of the extension and the distance from P to the building. 2. **Define variables:** Let: - $x$ = distance from point P to the building base (in meters) - $h$ = height of the building (in meters) - $e$ = height of the extension (in meters) 3. **Use tangent of angles:** - From P, the angle to the bottom of the extension (top of building) is 29°, so: $$\tan 29^\circ = \frac{h}{x}$$ - From P, the angle to the top of the extension is 32°, so: $$\tan 32^\circ = \frac{h + e}{x}$$ - From Q, which is 53 m closer, distance is $x - 53$, angle to top is 36°, so: $$\tan 36^\circ = \frac{h + e}{x - 53}$$ 4. **Express $h$ and $h+e$ from P's equations:** $$h = x \tan 29^\circ$$ $$h + e = x \tan 32^\circ$$ 5. **Find $e$ in terms of $x$:** $$e = (h + e) - h = x \tan 32^\circ - x \tan 29^\circ = x (\tan 32^\circ - \tan 29^\circ)$$ 6. **Use Q's equation:** $$\tan 36^\circ = \frac{h + e}{x - 53} = \frac{x \tan 32^\circ}{x - 53}$$ 7. **Solve for $x$:** $$\tan 36^\circ (x - 53) = x \tan 32^\circ$$ $$x \tan 36^\circ - 53 \tan 36^\circ = x \tan 32^\circ$$ $$x \tan 36^\circ - x \tan 32^\circ = 53 \tan 36^\circ$$ $$x (\tan 36^\circ - \tan 32^\circ) = 53 \tan 36^\circ$$ $$x = \frac{53 \tan 36^\circ}{\tan 36^\circ - \tan 32^\circ}$$ 8. **Calculate numerical values:** $$\tan 29^\circ \approx 0.5543$$ $$\tan 32^\circ \approx 0.6249$$ $$\tan 36^\circ \approx 0.7265$$ $$x = \frac{53 \times 0.7265}{0.7265 - 0.6249} = \frac{38.4845}{0.1016} \approx 378.9 \text{ m}$$ 9. **Calculate $h$ and $e$:** $$h = 378.9 \times 0.5543 \approx 210.0 \text{ m}$$ $$e = 378.9 \times (0.6249 - 0.5543) = 378.9 \times 0.0706 \approx 26.7 \text{ m}$$ **Final answers:** - Distance from P to building base: approximately 379 m - Height of building: approximately 210 m - Height of extension: approximately 27 m