1. **Problem statement:** Cybil and Fred are on opposite sides of a 200-foot-wide canyon. Both see the trail guide at an angle of depression of 60°. We need to find how far each is from the trail guide. 2. **Formula and rules:** The angle of depression equals the angle of elevation from the trail guide to each person. We can model the situation as two right triangles sharing the vertical canyon width of 200 feet. 3. **Step-by-step solution:** - Let the distance from Cybil to the trail guide be $x$ feet. - The vertical height (opposite side) is 200 feet. - The angle of depression (angle of elevation) is 60°. Using the tangent function: $$\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{200}{x}$$ Rearranging for $x$: $$x = \frac{200}{\tan(60^\circ)}$$ Since $\tan(60^\circ) = \sqrt{3}$: $$x = \frac{200}{\sqrt{3}}$$ Simplify by rationalizing the denominator: $$x = \frac{200}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{200\sqrt{3}}{3}$$ 4. **Final answer:** $$\boxed{\frac{200\sqrt{3}}{3} \approx 115.47 \text{ feet}}$$ So, Cybil and Fred are each approximately 115.47 feet from the trail guide.