1. **Problem statement:** We have a ski lift pole on a slope inclined at $21^\circ$ to the east. The shadow of the pole on the slope is 24 meters long. Sunlight comes from the west at an angle of $33^\circ$ to the horizontal. We need to find the height of the pole. 2. **Understanding the problem:** The pole stands perpendicular to the slope, which is inclined at $21^\circ$. The shadow length is measured along the slope (the hypotenuse of a right triangle formed by the pole, its shadow, and the vertical height). 3. **Key angles and setup:** - The slope angle: $21^\circ$ - Sunlight angle with horizontal: $33^\circ$ - Shadow length along slope: 24 m 4. **Step 1: Find the angle of sunlight relative to the slope surface.** Since the slope is inclined $21^\circ$ and sunlight is $33^\circ$ above horizontal, the angle between sunlight and slope surface is: $$\theta = 33^\circ - 21^\circ = 12^\circ$$ 5. **Step 2: Use the right triangle formed by the pole, its shadow, and the sunlight.** The shadow length $L = 24$ m is the adjacent side to angle $\theta = 12^\circ$ in the triangle where the pole height $h$ is opposite side. 6. **Step 3: Use the tangent function:** $$\tan(\theta) = \frac{h}{L}$$ 7. **Step 4: Solve for $h$:** $$h = L \times \tan(12^\circ)$$ 8. **Step 5: Calculate $h$ numerically:** $$h = 24 \times \tan(12^\circ) \approx 24 \times 0.2126 = 5.1024$$ 9. **Answer:** The height of the ski lift pole is approximately $5.1$ meters.