1. **Problem statement:** We have two tracking stations A and B, 49 miles apart. A satellite is above the ground at point C. The angles of elevation from A and B to the satellite are 87° and 84°, respectively. We need to find: (a) The distance from station A to the satellite (AC). (b) The height of the satellite above the ground (the vertical distance from C to line AB). 2. **Setup and formula:** We model the situation as triangle ABC with base AB = 49 miles. Angles at A and B are given: \(\angle A = 87^\circ\), \(\angle B = 84^\circ\). The angle at C is \(\angle C = 180^\circ - 87^\circ - 84^\circ = 9^\circ\). We use the Law of Sines: $$\frac{AC}{\sin B} = \frac{BC}{\sin A} = \frac{AB}{\sin C}$$ 3. **Calculate side AC (distance from A to satellite):** $$AC = \frac{AB \cdot \sin B}{\sin C} = \frac{49 \cdot \sin 84^\circ}{\sin 9^\circ}$$ Calculate the sines: $$\sin 84^\circ \approx 0.9945, \quad \sin 9^\circ \approx 0.1564$$ So, $$AC \approx \frac{49 \times 0.9945}{0.1564} \approx \frac{48.73}{0.1564} \approx 311.6 \text{ miles}$$ 4. **Calculate side BC (distance from B to satellite):** $$BC = \frac{AB \cdot \sin A}{\sin C} = \frac{49 \cdot \sin 87^\circ}{\sin 9^\circ}$$ Calculate the sine: $$\sin 87^\circ \approx 0.9986$$ So, $$BC \approx \frac{49 \times 0.9986}{0.1564} \approx \frac{48.93}{0.1564} \approx 312.9 \text{ miles}$$ 5. **Calculate the height of the satellite (altitude from C to AB):** The height is the altitude from C to AB. Using the formula for area of triangle: $$\text{Area} = \frac{1}{2} \times AB \times \text{height}$$ Also, $$\text{Area} = \frac{1}{2} AC \times BC \times \sin C$$ Equate and solve for height: $$\frac{1}{2} AB \times h = \frac{1}{2} AC \times BC \times \sin C$$ $$h = \frac{AC \times BC \times \sin C}{AB}$$ Substitute values: $$h = \frac{311.6 \times 312.9 \times 0.1564}{49}$$ Calculate numerator: $$311.6 \times 312.9 \approx 97588.6$$ Then, $$h \approx \frac{97588.6 \times 0.1564}{49} = \frac{15258.5}{49} \approx 311.2 \text{ miles}$$ **Final answers:** (a) Distance from station A to satellite: **311.6 miles** (b) Height of the satellite above the ground: **311.2 miles**