1. **Problem statement:** Lara stands 40 meters from a tree and observes the tree's top at a 30° angle of elevation and the bottom of its reflection in the water at a 45° angle of depression. We need to find expressions for the tree's height, the length of its reflection, and the total vertical distance between the top of the tree and the bottom of its reflection. 2. **Formulas and rules:** - For angle of elevation, height $h = d \tan(\theta)$ where $d$ is horizontal distance and $\theta$ is the angle of elevation. - For angle of depression, the length of reflection $r = d \tan(\phi)$ where $\phi$ is the angle of depression. - The total vertical distance is the sum of the tree's height and the reflection length. 3. **Calculate the height of the tree:** $$h = 40 \times \tan(30^\circ)$$ Using $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, $$h = 40 \times \frac{1}{\sqrt{3}} = \frac{40}{\sqrt{3}}$$ 4. **Calculate the length of the tree's reflection:** $$r = 40 \times \tan(45^\circ)$$ Since $\tan(45^\circ) = 1$, $$r = 40 \times 1 = 40$$ 5. **Calculate the total vertical distance:** $$\text{Total distance} = h + r = \frac{40}{\sqrt{3}} + 40$$ **Final answers:** - Height of the tree: $h = \frac{40}{\sqrt{3}}$ meters - Length of the reflection: $r = 40$ meters - Total vertical distance: $\frac{40}{\sqrt{3}} + 40$ meters