1. **Problem Statement:** We have a lighthouse 350 feet above sea level. Two angles of depression from the lighthouse to two ships offshore are given as $\theta$ (upper angle) and $\beta$ (lower angle). We want to understand the relationship between these angles and the distances to the ships. 2. **Understanding Angles of Depression:** The angle of depression is the angle between the horizontal line from the observer's eye (top of the lighthouse) and the line of sight to the object (ship). By geometry, the angle of depression equals the angle of elevation from the ship to the lighthouse base. 3. **Using Trigonometry:** Let the horizontal distances from the lighthouse base to the ships be $x_\theta$ for the ship at angle $\theta$ and $x_\beta$ for the ship at angle $\beta$. 4. **Formulas:** Using tangent, which relates opposite side (height) to adjacent side (distance): $$\tan(\theta) = \frac{350}{x_\theta} \quad \Rightarrow \quad x_\theta = \frac{350}{\tan(\theta)}$$ $$\tan(\beta) = \frac{350}{x_\beta} \quad \Rightarrow \quad x_\beta = \frac{350}{\tan(\beta)}$$ 5. **Interpretation:** - The ship with angle $\theta$ is closer if $\theta > \beta$ because $\tan(\theta)$ is larger, making $x_\theta$ smaller. - The ship with angle $\beta$ is farther offshore. 6. **Summary:** To find the distances to the ships offshore, use the formulas: $$x_\theta = \frac{350}{\tan(\theta)}$$ $$x_\beta = \frac{350}{\tan(\beta)}$$ This completes the analysis of the problem.