1. **State the problem:** Romeo and Paris are observing Juliet's balcony from two different positions. Romeo faces north and sees the balcony at an angle of elevation of 20 degrees. Paris faces west and sees the balcony at an angle of elevation of 18 degrees. Romeo and Paris are 100 m apart. We need to find the height of Juliet's balcony above the ground. 2. **Set up the scenario:** Let the height of the balcony be $h$ meters. Let the horizontal distances from Romeo and Paris to the base of the balcony be $d_R$ and $d_P$ respectively. 3. **Use the tangent of the angle of elevation:** - For Romeo: $\tan(20^\circ) = \frac{h}{d_R}$ so $h = d_R \tan(20^\circ)$ - For Paris: $\tan(18^\circ) = \frac{h}{d_P}$ so $h = d_P \tan(18^\circ)$ 4. **Relate the distances:** Romeo and Paris are 100 m apart, and since Romeo faces north and Paris faces west, their positions and the balcony form a right triangle with legs $d_R$ and $d_P$ and hypotenuse 100 m. 5. **Apply the Pythagorean theorem:** $$d_R^2 + d_P^2 = 100^2 = 10000$$ 6. **Express $d_P$ in terms of $d_R$ using the height equations:** Since $h = d_R \tan(20^\circ) = d_P \tan(18^\circ)$, $$d_P = d_R \frac{\tan(20^\circ)}{\tan(18^\circ)}$$ 7. **Substitute $d_P$ into the Pythagorean theorem:** $$d_R^2 + \left(d_R \frac{\tan(20^\circ)}{\tan(18^\circ)}\right)^2 = 10000$$ $$d_R^2 \left(1 + \left(\frac{\tan(20^\circ)}{\tan(18^\circ)}\right)^2\right) = 10000$$ 8. **Calculate the tangent values:** $$\tan(20^\circ) \approx 0.3640$$ $$\tan(18^\circ) \approx 0.3249$$ 9. **Calculate the ratio and simplify:** $$\frac{\tan(20^\circ)}{\tan(18^\circ)} \approx \frac{0.3640}{0.3249} \approx 1.120$$ 10. **Calculate the sum inside the parentheses:** $$1 + (1.120)^2 = 1 + 1.2544 = 2.2544$$ 11. **Solve for $d_R^2$:** $$d_R^2 = \frac{10000}{2.2544} \approx 4434.5$$ 12. **Find $d_R$:** $$d_R = \sqrt{4434.5} \approx 66.6 \text{ m}$$ 13. **Find $h$ using Romeo's angle:** $$h = d_R \tan(20^\circ) = 66.6 \times 0.3640 \approx 24.2 \text{ m}$$ 14. **Round to the nearest meter:** $$h \approx 24 \text{ meters}$$ **Final answer:** The height of Juliet's balcony above the ground is approximately 24 meters.