1. **Problem statement:** A man at the top of a 75 m high tower sees two goats due west at angles of depression 10° and 17°. We need to find the distance between the two goats. 2. **Formula and concepts:** The angle of depression from the top of the tower to each goat forms a right triangle with the tower height as the opposite side and the horizontal distance to each goat as the adjacent side. Using the tangent function: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ Here, opposite = 75 m (height of tower), adjacent = distance to goat. 3. **Calculate distances:** - For the goat at 10° angle of depression: $$d_1 = \frac{75}{\tan(10^\circ)}$$ - For the goat at 17° angle of depression: $$d_2 = \frac{75}{\tan(17^\circ)}$$ 4. **Evaluate distances:** - Calculate $\tan(10^\circ) \approx 0.1763$, so $$d_1 = \frac{75}{0.1763} \approx 425.5 \text{ m}$$ - Calculate $\tan(17^\circ) \approx 0.3057$, so $$d_2 = \frac{75}{0.3057} \approx 245.4 \text{ m}$$ 5. **Find distance between goats:** Since both goats are due west, the distance between them is the difference of their distances from the tower: $$\text{Distance} = d_1 - d_2 = 425.5 - 245.4 = 180.1 \text{ m}$$ **Final answer:** The distance between the two goats is approximately **180.1 meters**.