1. **State the problem:** We have a viewing tower 30 meters above the ground. The angle of depression to an object on the ground is 30 degrees, and the angle of elevation to an aircraft vertically above that object is 42 degrees. We need to find the height of the aircraft above the ground. 2. **Define variables and draw a diagram:** Let the horizontal distance from the base of the tower to the object be $x$ meters. Let the height of the aircraft above the ground be $h$ meters. 3. **Use the angle of depression to find $x$:** The angle of depression from the tower to the object is 30 degrees. Since the tower height is 30 meters, by right triangle trigonometry: $$\tan(30^\circ) = \frac{30}{x}$$ So, $$x = \frac{30}{\tan(30^\circ)}$$ We know $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, so $$x = 30 \times \sqrt{3} \approx 30 \times 1.732 = 51.96 \text{ meters}$$ 4. **Use the angle of elevation to find the height of the aircraft above the object:** The aircraft is vertically above the object, so the horizontal distance from the tower to the aircraft is also $x$. The angle of elevation from the tower to the aircraft is 42 degrees. Let the height of the aircraft above the tower be $h_a$. By right triangle trigonometry: $$\tan(42^\circ) = \frac{h_a}{x}$$ So, $$h_a = x \times \tan(42^\circ)$$ Using $\tan(42^\circ) \approx 0.9004$, $$h_a = 51.96 \times 0.9004 \approx 46.79 \text{ meters}$$ 5. **Calculate the total height of the aircraft above the ground:** The tower height is 30 meters, so $$h = 30 + h_a = 30 + 46.79 = 76.79 \text{ meters}$$ **Final answer:** The height of the aircraft above the ground is approximately **76.79 meters**.