1. **Problem Statement:** From the top of a rock 100 m high, the depression angles to the top and base of a tower are 22° and 33° respectively. The base of the rock and the tower are on the same horizontal plane. Find the height of the tower. 2. **Diagram Explanation:** Imagine a vertical rock 100 m tall. From its top, two lines of sight go downwards: one at 22° depression to the tower's top, and another at 33° depression to the tower's base. The horizontal distance from the rock base to the tower base is the same for both lines. 3. **Formulas and Rules:** - Depression angle equals the angle between the horizontal line from the observer and the line of sight downward. - Use right triangle trigonometry: if $\theta$ is the depression angle and $d$ is horizontal distance, then vertical height difference $h = d \tan \theta$. 4. **Step-by-step Solution:** - Let $d$ be the horizontal distance from the rock base to the tower base. - From the top of the rock, the line of sight to the tower base forms a right triangle with angle 33°: $$\tan 33^\circ = \frac{100}{d} \implies d = \frac{100}{\tan 33^\circ}$$ - Calculate $d$: $$d = \frac{100}{\tan 33^\circ} \approx \frac{100}{0.6494} \approx 153.9\,m$$ - From the top of the rock to the tower top, angle is 22°: $$\tan 22^\circ = \frac{100 - h}{d} \implies 100 - h = d \tan 22^\circ$$ - Calculate $100 - h$: $$100 - h = 153.9 \times \tan 22^\circ \approx 153.9 \times 0.4040 = 62.2$$ - Solve for $h$: $$h = 100 - 62.2 = 37.8\,m$$ 5. **Answer:** The height of the tower is approximately **38 meters**.