1. **Problem statement:** Two planes are flying at an altitude of 2.5 miles approaching O’Hare Airport from opposite directions. Flight 104 sees the tower at an angle of depression of 17.78°, and flight 217 sees it at 12.65°. We need to find the distance between the two planes. 2. **Setup:** Let the horizontal distances from each plane to the tower be $d_1$ and $d_2$ respectively. The total distance between the planes is $d_1 + d_2$. 3. **Using the angle of depression:** The altitude forms the opposite side of a right triangle, and the horizontal distance is the adjacent side. Using the tangent function: $$\tan(17.78^\circ) = \frac{2.5}{d_1} \implies d_1 = \frac{2.5}{\tan(17.78^\circ)}$$ $$\tan(12.65^\circ) = \frac{2.5}{d_2} \implies d_2 = \frac{2.5}{\tan(12.65^\circ)}$$ 4. **Calculate $d_1$ and $d_2$:** $$d_1 = \frac{2.5}{\tan(17.78^\circ)}$$ $$d_2 = \frac{2.5}{\tan(12.65^\circ)}$$ 5. **Distance between planes:** $$D = d_1 + d_2 = \frac{2.5}{\tan(17.78^\circ)} + \frac{2.5}{\tan(12.65^\circ)}$$ 6. **Numerical evaluation:** Calculate each tangent: $$\tan(17.78^\circ) \approx 0.321$$ $$\tan(12.65^\circ) \approx 0.225$$ Then: $$d_1 = \frac{2.5}{0.321} \approx 7.79 \text{ miles}$$ $$d_2 = \frac{2.5}{0.225} \approx 11.11 \text{ miles}$$ 7. **Final answer:** $$D = 7.79 + 11.11 = 18.90 \text{ miles}$$ The distance between the two planes is approximately 18.90 miles.