1. **Problem statement:** We have two tracking stations A and B on the ground, 50 miles apart horizontally. The angles of elevation to a satellite from A and B are 87.0° and 84.2°, respectively. We want to find the height of the satellite above the ground. 2. **Formula and approach:** Let the height of the satellite be $h$ miles, and let the horizontal distance from station A to the satellite's ground projection be $x$ miles. Then the distance from B to the satellite's ground projection is $50 - x$ miles. Using the tangent of the angles of elevation: $$\tan(87.0^\circ) = \frac{h}{x}$$ $$\tan(84.2^\circ) = \frac{h}{50 - x}$$ 3. **Set up equations:** From the first, $$h = x \tan(87.0^\circ)$$ From the second, $$h = (50 - x) \tan(84.2^\circ)$$ 4. **Equate and solve for $x$:** $$x \tan(87.0^\circ) = (50 - x) \tan(84.2^\circ)$$ $$x \tan(87.0^\circ) = 50 \tan(84.2^\circ) - x \tan(84.2^\circ)$$ $$x (\tan(87.0^\circ) + \tan(84.2^\circ)) = 50 \tan(84.2^\circ)$$ $$x = \frac{50 \tan(84.2^\circ)}{\tan(87.0^\circ) + \tan(84.2^\circ)}$$ 5. **Calculate values:** $$\tan(87.0^\circ) \approx 19.081$$ $$\tan(84.2^\circ) \approx 9.514$$ So, $$x = \frac{50 \times 9.514}{19.081 + 9.514} = \frac{475.7}{28.595} \approx 16.64$$ 6. **Find height $h$:** $$h = x \tan(87.0^\circ) = 16.64 \times 19.081 \approx 317.5$$ **Final answer:** The height of the satellite above the ground is approximately **317.5 miles**.