A wall is to be supported by a wooden pole. In order for a support pole to keep a wall vertical, the angle of elevation of the pole must be within the range of \( 30-60 \) degrees. If the pole is 8 feet long, determine the distance from the base of the wall to position the pole.

To determine the appropriate distance from the base of the wall to position the 8-foot-long support pole, we can use trigonometry, specifically the cosine function. The angle of elevation (\(\theta\)) of the pole affects the horizontal distance (\(d\)) from the wall. The relationship is given by: \[ d = \text{length of pole} \times \cos(\theta) \] Given: - Length of the pole (\(L\)) = 8 feet - Angle of elevation (\(\theta\)) varies between 30° and 60° Let's calculate the distance for both extreme angles. 1. **At \(\theta = 60^\circ\):** \[ d = 8 \times \cos(60^\circ) = 8 \times 0.5 = 4 \text{ feet} \] 2. **At \(\theta = 30^\circ\):** \[ d = 8 \times \cos(30^\circ) = 8 \times \left(\frac{\sqrt{3}}{2}\right) = 4\sqrt{3} \text{ feet} \approx 6.93 \text{ feet} \] **Conclusion:** To ensure the wall remains vertical, the base of the pole should be placed at a distance between **4 feet** and **\(4\sqrt{3}\) feet** (approximately 6.93 feet) from the wall. **Final Answer:** Place the pole so its base is between 4 feet and 4 √3 feet from the wall.