1. **State the problem:** We have an observatory 150 feet high and an angle of depression of 25° to an island. We want to find the horizontal distance from the observatory to the island. 2. **Identify the right triangle and angle:** The height of the observatory is the vertical side (opposite the angle of depression), and the horizontal distance to the island is the adjacent side to the 25° angle. 3. **Use the tangent function:** Tangent relates the opposite side to the adjacent side in a right triangle: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ Here, $\theta = 25^\circ$, opposite = 150 ft, adjacent = horizontal distance $d$. 4. **Set up the equation:** $$\tan(25^\circ) = \frac{150}{d}$$ 5. **Solve for $d$:** $$d = \frac{150}{\tan(25^\circ)}$$ 6. **Calculate $\tan(25^\circ)$:** $$\tan(25^\circ) \approx 0.4663$$ 7. **Substitute and compute:** $$d = \frac{150}{0.4663} \approx 321.8$$ 8. **Interpret the result:** The horizontal distance is approximately 322 feet. **Final answer:** C. 322 feet