1. **Problem statement:** Romeo and Paris are observing Juliet's balcony from two points 100 m apart. Romeo sees the balcony at an angle of elevation of 20° facing north, and Paris sees it at 18° facing west. We need to find the height $h$ of Juliet's balcony above the ground. 2. **Setup and formula:** Let $R$ be Romeo's position, $P$ be Paris's position, and $J$ be the balcony top. The points $R$, $P$, and $J$ form a triangle with base $RP = 100$ m. The angles of elevation at $R$ and $P$ are $20^\circ$ and $18^\circ$ respectively. 3. **Key idea:** The height $h$ is the vertical leg of two right triangles $R-J$ and $P-J$ sharing the same height $h$. We use the tangent function which relates angle of elevation to height and horizontal distance: $$\tan(\theta) = \frac{\text{height}}{\text{horizontal distance}}$$ 4. **Define variables:** Let $x$ be the horizontal distance from Romeo to the point on the ground directly below the balcony $J$. Then the distance from Paris to that point is $100 - x$. 5. **Write equations using tangent:** From Romeo's perspective: $$\tan(20^\circ) = \frac{h}{x} \implies h = x \tan(20^\circ)$$ From Paris's perspective: $$\tan(18^\circ) = \frac{h}{100 - x} \implies h = (100 - x) \tan(18^\circ)$$ 6. **Set the two expressions for $h$ equal:** $$x \tan(20^\circ) = (100 - x) \tan(18^\circ)$$ 7. **Solve for $x$:** $$x \tan(20^\circ) = 100 \tan(18^\circ) - x \tan(18^\circ)$$ $$x (\tan(20^\circ) + \tan(18^\circ)) = 100 \tan(18^\circ)$$ $$x = \frac{100 \tan(18^\circ)}{\tan(20^\circ) + \tan(18^\circ)}$$ 8. **Calculate numerical values:** $$\tan(20^\circ) \approx 0.36397$$ $$\tan(18^\circ) \approx 0.32492$$ $$x = \frac{100 \times 0.32492}{0.36397 + 0.32492} = \frac{32.492}{0.68889} \approx 47.17 \text{ m}$$ 9. **Calculate height $h$:** $$h = x \tan(20^\circ) = 47.17 \times 0.36397 \approx 17.16 \text{ m}$$ 10. **Final answer:** The height of Juliet's balcony is approximately **17 metres**.